The radius of convexity of three kind of normalized Bessel functions of the first kind1 Árpád Baricz2 Babes-Bolyai ¸ University, Department of Economics, Cluj-Napoca, Romania Óbuda University, John von Neumann Faculty of Informatics, Institute of Applied Mathematics, Budapest, Hungary https://sites.google.com/site/bariczocsi/, e-mail: [email protected] Mathematics and Statistics Seminar Department of Mathematics and Statistics University of Cyprus Nicosia, Cyprus

1

Based on the papers arXiv.1202.1504 and arxiv.1302.4222. Supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2012-3-0190. 2

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

1 / 35

Abstract In this talk our aim is to determine the radius of convexity for three kind of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

2 / 35

Definition of starlike and convex functions

I Let D(z0 , r ) = {z ∈ C : |z − z0 | < r } be the open disk with center z0 ∈ C and radius r > 0 and let us denote the particular disk D(0, 1) by D. Moreover, let A be the class of analytic and univalent functions defined in the unit disk D, which can be normalized as f (z) = z + a2 z 2 + a3 z 3 + . . . , that is, f (0) = f 0 (0) − 1 = 0. The class of starlike functions, denoted by S ∗ , is the subclass of A which consists of functions f for which the domain f (D) is starlike with respect to 0. Recall also that a function f ∈ A belongs to the class K of convex functions if the image domain f (D) is a convex domain in C, that is, the domain f (D) ⊂ C contains the entire line segment joining any pair of its points.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

3 / 35

Definition of starlike and convex functions

I Let D(z0 , r ) = {z ∈ C : |z − z0 | < r } be the open disk with center z0 ∈ C and radius r > 0 and let us denote the particular disk D(0, 1) by D. Moreover, let A be the class of analytic and univalent functions defined in the unit disk D, which can be normalized as f (z) = z + a2 z 2 + a3 z 3 + . . . , that is, f (0) = f 0 (0) − 1 = 0. The class of starlike functions, denoted by S ∗ , is the subclass of A which consists of functions f for which the domain f (D) is starlike with respect to 0. Recall also that a function f ∈ A belongs to the class K of convex functions if the image domain f (D) is a convex domain in C, that is, the domain f (D) ⊂ C contains the entire line segment joining any pair of its points.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

3 / 35

Characterization of starlike and convex functions I An analytic description of S ∗ is  0    zf (z) ∗ S = f ∈ A Re > 0 for all z ∈ D . f (z) Moreover, consider the class of starlike functions of order β ∈ [0, 1), that is,  0    zf (z) ∗ > β for all z ∈ D . S (β) = f ∈ A Re f (z)

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

4 / 35

Characterization of starlike and convex functions I An analytic description of S ∗ is  0    zf (z) ∗ S = f ∈ A Re > 0 for all z ∈ D . f (z) Moreover, consider the class of starlike functions of order β ∈ [0, 1), that is,  0    zf (z) ∗ > β for all z ∈ D . S (β) = f ∈ A Re f (z) I It is also known that the class of convex functions can be characterized as     zf 00 (z) K = f ∈ A Re 1 + 0 > 0, z ∈ D . f (z) Moreover, for α ∈ [0, 1) we consider also the class of convex functions of order α defined by     zf 00 (z) > α, z ∈ D . K(α) = f ∈ A Re 1 + 0 f (z)

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

4 / 35

Definitions of radii of starlikeness and convexity I The real numbers  0    zf (z) r (f ) = sup r > 0 Re > 0 for all z ∈ D(0, r ) f (z) ∗

and

 0    zf (z) rβ∗ (f ) = sup r > 0 Re > β for all z ∈ D(0, r ) , f (z)

are called the radius of starlikeness and the radius of starlikeness of order β of the function f , respectively. We note that in fact r ∗ (f ) is the largest radius such that f (D(0, r ∗ (f ))) is a starlike domain with respect to 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

5 / 35

Definitions of radii of starlikeness and convexity I The real numbers  0    zf (z) r (f ) = sup r > 0 Re > 0 for all z ∈ D(0, r ) f (z) ∗

and

 0    zf (z) rβ∗ (f ) = sup r > 0 Re > β for all z ∈ D(0, r ) , f (z)

are called the radius of starlikeness and the radius of starlikeness of order β of the function f , respectively. We note that in fact r ∗ (f ) is the largest radius such that f (D(0, r ∗ (f ))) is a starlike domain with respect to 0. I Now let us consider the radius of convexity, and the radius of convexity of order α of the function f     zf 00 (z) r c (f ) = sup r > 0 Re 1 + 0 > 0, z ∈ D(0, r ) f (z) and

    zf 00 (z) rαc (f ) = sup r > 0 Re 1 + 0 > α, z ∈ D(0, r ) . f (z)

We note that r c (f ) is in fact the largest radius for which the image domain f (D(0, r c (f ))) is a convex domain in C. Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

5 / 35

Normalized Bessel functions of the first kind I Now, consider the Bessel function of the first kind, which is a particular solution of the second-order linear homogeneous Bessel differential equation. This function has the infinite series representation  z 2n+ν X (−1)n Jν (z) = , n!Γ(n + ν + 1) 2 n≥0

where z ∈ C and ν ∈ C such that ν 6= −1, −2, . . .. Observe that the Bessel function Jν does not belong to class A. Thus, it is natural to consider the following three kind of normalization of the Bessel function of the first kind 1 fν (z) = [2ν Γ(ν + 1)Jν (z)]1/ν = z − z 3 + . . . , ν 6= 0, (1) 4ν(ν + 1) gν (z) = 2ν Γ(ν + 1)z 1−ν Jν (z) = z − and

1 1 z3 + z5 − . . . 4(ν + 1) 32(ν + 1)(ν + 2)

(2)

√ hν (z) = 2ν Γ(ν + 1)z 1−ν/2 Jν ( z) = z −

1 z2 + . . . . (3) 4(ν + 1) Clearly the functions fν , gν and hν belong to the class A. We note here that in fact   1 fν (z) = exp Log (2ν Γ(ν + 1)Jν (z)) , ν where Log represents the principal branch of the logarithm function and every many-valued function are taken with the principal branch. Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

6 / 35

Geometric properties of normalized Bessel functions I Now, let us recall some results on the geometric behavior of the functions fν , gν and hν . Brown [Br1] determined the radius of starlikeness for fν in the case when ν > 0. Namely, in [Br1, Theorem 2] it was shown that the radius r ∗ (fν ) is the smallest positive zero of the function z 7→ Jν0 (z). Moreover, in [Br1, Theorem 3] Brown proved that if ν > 0, then the radius of starlikeness of the function gν is the smallest positive zero of the function z 7→ zJν0 (z) + (1 − ν)Jν (z). Kreyszig and Todd [KT, Theorem 3] proved that when ν > −1 the function gν is univalent in the circle |z| ≤ ρν but not in any concentric circle with larger radius, where ρν is the first maximum of the function gν on the positive real axis. Brown [Br1, p. 282] pointed out that when ν > 0 the radius of starlikeness of the function gν , that is, r ∗ (gν ) is exactly the radius of univalence ρν obtained by Kreyszig and Todd [KT]. Furthermore, Brown [Br2, Theorem 5.1] showed that the radius of starlikeness of the function gν is also ρν when ν ∈ (−1/2, 0). On the other hand, Hayden and Merkes [HM, Theorem C] deduced that when µ = Re ν > −1 the radius of starlikeness of gν is not less than the smallest positive zero of gµ0 . It is worth to mention that Brown used the methods of Nehari [Ne] and Robertson [Ro], and an important tool in the proofs was the fact that the Bessel function of the first kind is a particular solution of the Bessel differential equation.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

7 / 35

The radius of starlikeness of normalized Bessel functions

Theorem Let 1 > β ≥ 0. Then the following assertions are true: a. If ν ∈ (−1, 0), then rβ∗ (fν ) = xν,β , where xν,β denotes the unique positive root of the equation zIν0 (z) − βνIν (z) = 0. Moreover, if ν > 0, then we have rβ∗ (fν ) = xν,β,1 , where xν,β,1 is the smallest positive root of the equation zJν0 (z) − βνJν (z) = 0. b. If ν > −1, then rβ∗ (gν ) = yν,β,1 , where yν,β,1 is the smallest positive root of the equation zJν0 (z) + (1 − β − ν)Jν (z) = 0. c. If ν > −1, then rβ∗ (hν ) = zν,β,1 , where zν,β,1 is the smallest positive root of the equation zJν0 (z) + (2 − 2β − ν)Jν (z) = 0. Here Iν denotes the modified Bessel function of the first kind, which in view of the relation Iν (z) = i−ν Jν (iz) is also called sometimes as the Bessel function of the first kind with imaginary argument.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

8 / 35

Corollary The following assertions are true: a. If ν ∈ (−1, 0), then the radius of starlikeness of fν is xν,0 , where xν,0 is the unique positive root of the equation Iν0 (z) = 0. If ν > 0, then the radius of starlikeness of the function fν is xν,0,1 , which denotes the smallest positive root of the equation Jν0 (z) = 0. b. If ν > −1, then the radius of starlikeness of the function gν is yν,0,1 , which denotes the smallest positive root of the equation zJν0 (z) + (1 − ν)Jν (z) = 0. c. If ν > −1, then the radius of starlikeness of the function hν is zν,0,1 , which denotes the smallest positive root of the equation zJν0 (z) + (2 − ν)Jν (z) = 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

9 / 35

Corollary The following assertions are true: a. If ν ∈ (−1, 0), then the radius of starlikeness of fν is xν,0 , where xν,0 is the unique positive root of the equation Iν0 (z) = 0. If ν > 0, then the radius of starlikeness of the function fν is xν,0,1 , which denotes the smallest positive root of the equation Jν0 (z) = 0. b. If ν > −1, then the radius of starlikeness of the function gν is yν,0,1 , which denotes the smallest positive root of the equation zJν0 (z) + (1 − ν)Jν (z) = 0. c. If ν > −1, then the radius of starlikeness of the function hν is zν,0,1 , which denotes the smallest positive root of the equation zJν0 (z) + (2 − ν)Jν (z) = 0. I Observe that part a and b of Corollary 2 complement the results of [Br1, Theorem 2], [Br1, Theorem 3] and [Br2, Theorem 5.1], mentioned above. It is of interest to note here that very recently Szász [Sz] proved that the normalized Bessel function hν is starlike if and only if ν ≥ ν0 , where ν0 = −0.5623 . . . is the root of the equation hν0 (1) = 0, that is, Jν0 (1) + (2 − ν)Jν (1) = 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

9 / 35

Proof I First we prove part a for ν > 0 and parts b and c for ν > −1. We need to show that the inequalities  0   0   0  zfν (z) zgν (z) zhν (z) Re > β, Re > β and Re >β (4) fν (z) gν (z) hν (z) are valid for all ν > 0 and z ∈ D(0, xν,β,1 ), ν > −1 and z ∈ D(0, yν,β,1 ), and ν > −1 and z ∈ D(0, zν,β,1 ), respectively, and each of the above inequalities does not hold in any larger disk.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

10 / 35

Proof I First we prove part a for ν > 0 and parts b and c for ν > −1. We need to show that the inequalities  0   0   0  zfν (z) zgν (z) zhν (z) Re > β, Re > β and Re >β (4) fν (z) gν (z) hν (z) are valid for all ν > 0 and z ∈ D(0, xν,β,1 ), ν > −1 and z ∈ D(0, yν,β,1 ), and ν > −1 and z ∈ D(0, zν,β,1 ), respectively, and each of the above inequalities does not hold in any larger disk. I Lommel’s well-known result states that if ν > −1, then the zeros of the Bessel function Jν are all real. Thus, if jν,n denotes the n-th positive zero of the Bessel function Jν , then the Bessel function admits the Weierstrassian decomposition of the form [Wa, p. 498]  Y z2 zν Jν (z) = ν 1− 2 , (5) 2 Γ(ν + 1) jν,n n≥1 and this infinite product is uniformly convergent on each compact subset of C.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

10 / 35

Proof I Logarithmic differentiation of (5) yields X 2z 2 zJν0 (z) , =ν− Jν (z) j2 − z2 n≥1 ν,n

(6)

which in view of the recurrence relation [Wa, p. 45] zJν0 (z) − νJν (z) = −zJν+1 (z) is equivalent to the Mittag-Leffler expansion [Wa, p. 498] X Jν+1 (z) 2z . = 2 Jν (z) j − z2 ν,n n≥1

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

11 / 35

Proof I Logarithmic differentiation of (5) yields X 2z 2 zJν0 (z) , =ν− Jν (z) j2 − z2 n≥1 ν,n

(6)

which in view of the recurrence relation [Wa, p. 45] zJν0 (z) − νJν (z) = −zJν+1 (z) is equivalent to the Mittag-Leffler expansion [Wa, p. 498] X Jν+1 (z) 2z . = 2 Jν (z) j − z2 ν,n n≥1 Consequently, in view of (1),(2), (3) and (6) we obtain zfν0 (z) 1 zJν0 (z) 1 X 2z 2 = =1− , fν (z) ν Jν (z) ν j2 − z2 n≥1 ν,n

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

11 / 35

Proof I Logarithmic differentiation of (5) yields X 2z 2 zJν0 (z) , =ν− Jν (z) j2 − z2 n≥1 ν,n

(6)

which in view of the recurrence relation [Wa, p. 45] zJν0 (z) − νJν (z) = −zJν+1 (z) is equivalent to the Mittag-Leffler expansion [Wa, p. 498] X Jν+1 (z) 2z . = 2 Jν (z) j − z2 ν,n n≥1 Consequently, in view of (1),(2), (3) and (6) we obtain zfν0 (z) 1 zJν0 (z) 1 X 2z 2 = =1− , fν (z) ν Jν (z) ν j2 − z2 n≥1 ν,n X 2z 2 zgν0 (z) zJ 0 (z) =1−ν+ ν =1− gν (z) Jν (z) j2 − z2 n≥1 ν,n

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

11 / 35

Proof I Logarithmic differentiation of (5) yields X 2z 2 zJν0 (z) , =ν− Jν (z) j2 − z2 n≥1 ν,n

(6)

which in view of the recurrence relation [Wa, p. 45] zJν0 (z) − νJν (z) = −zJν+1 (z) is equivalent to the Mittag-Leffler expansion [Wa, p. 498] X Jν+1 (z) 2z . = 2 Jν (z) j − z2 ν,n n≥1 Consequently, in view of (1),(2), (3) and (6) we obtain zfν0 (z) 1 zJν0 (z) 1 X 2z 2 = =1− , fν (z) ν Jν (z) ν j2 − z2 n≥1 ν,n X 2z 2 zgν0 (z) zJ 0 (z) =1−ν+ ν =1− gν (z) Jν (z) j2 − z2 n≥1 ν,n and

√ √ X zhν0 (z) ν 1 zJν0 ( z) z √ =1− + =1− . 2 hν (z) 2 2 Jν ( z) j −z n≥1 ν,n

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

11 / 35

Proof

I It is known [Wa, p. 597] that in case α + ν > 0 and ν > −1 the so-called Dini function z 7→ zJν0 (z) + αJν (z) has only real zeros and according to Ismail and Muldoon [IM, p. 11] we know that the smallest positive zero of the above function is less than jν,1 . This in turn implies that xν,β,1 < jν,1 for all ν > 0, yν,β,1 < jν,1 for all ν > −1, and zν,β,1 < jν,1 for all ν > −1. In other words, for all 0 ≤ β < 1 and n ∈ {2, 3, . . .} we have D(0, xν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > 0, D(0, yν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > −1, and D(0, zν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > −1.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

12 / 35

Proof

I It is known [Wa, p. 597] that in case α + ν > 0 and ν > −1 the so-called Dini function z 7→ zJν0 (z) + αJν (z) has only real zeros and according to Ismail and Muldoon [IM, p. 11] we know that the smallest positive zero of the above function is less than jν,1 . This in turn implies that xν,β,1 < jν,1 for all ν > 0, yν,β,1 < jν,1 for all ν > −1, and zν,β,1 < jν,1 for all ν > −1. In other words, for all 0 ≤ β < 1 and n ∈ {2, 3, . . .} we have D(0, xν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > 0, D(0, yν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > −1, and D(0, zν,β,1 ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν > −1. I On the other hand, it is known [Sz] that if z ∈ C and α ∈ R such that α > |z|, then   |z| z ≥ Re . (7) α − |z| α−z

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

12 / 35

Proof I By using (7), we obtain for all ν > −1, n ∈ {1, 2, . . . } and z ∈ D(0, jν,1 ) the inequality   |z|2 z2 ≥ Re , (8) 2 2 jν,n − |z|2 jν,n − z2 which in turn implies that    X 2z 2 X 2|z|2 zfν0 (z) |z|fν0 (|z|) 1 ≥1− 1 = 1 − Re  Re , = fν (z) ν ν fν (|z|) j2 − z2 j 2 − |z|2 n≥1 ν,n n≥1 ν,n 

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

13 / 35

Proof I By using (7), we obtain for all ν > −1, n ∈ {1, 2, . . . } and z ∈ D(0, jν,1 ) the inequality   |z|2 z2 ≥ Re , (8) 2 2 jν,n − |z|2 jν,n − z2 which in turn implies that    X 2z 2 X 2|z|2 zfν0 (z) |z|fν0 (|z|) 1 ≥1− 1 = 1 − Re  Re , = fν (z) ν ν fν (|z|) j2 − z2 j 2 − |z|2 n≥1 ν,n n≥1 ν,n 

   X 2z 2 X 2|z|2 zgν0 (z) |z|gν0 (|z|) ≥1− Re = 1 − Re  = 2 2 2 2 gν (z) gν (|z|) j −z j − |z| n≥1 ν,n n≥1 ν,n 

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

13 / 35

Proof I By using (7), we obtain for all ν > −1, n ∈ {1, 2, . . . } and z ∈ D(0, jν,1 ) the inequality   |z|2 z2 ≥ Re , (8) 2 2 jν,n − |z|2 jν,n − z2 which in turn implies that    X 2z 2 X 2|z|2 zfν0 (z) |z|fν0 (|z|) 1 ≥1− 1 = 1 − Re  Re , = fν (z) ν ν fν (|z|) j2 − z2 j 2 − |z|2 n≥1 ν,n n≥1 ν,n 

   X 2z 2 X 2|z|2 zgν0 (z) |z|gν0 (|z|) ≥1− Re = 1 − Re  = 2 2 2 2 gν (z) gν (|z|) j −z j − |z| n≥1 ν,n n≥1 ν,n 

and    X X |z|hν0 (|z|) zhν0 (z) |z| z ≥1− Re = 1 − Re  = , hν (z) hν (|z|) j2 − z j 2 − |z| n≥1 ν,n n≥1 ν,n 

with equality when z = |z| = r . Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

13 / 35

Proof

I The minimum principle for harmonic functions and the previous inequalities imply that the corresponding inequalities in (4) are valid if and only if we have |z| < xν,β,1 , |z| < yν,β,1 , and |z| < zν,β,1 , respectively, where xν,β,1 , yν,β,1 and zν,β,1 are the smallest positive roots of the equations rfν0 (r )/fν (r ) = β, rgν0 (r )/gν (r ) = β and rhν0 (r )/hν (r ) = β, which are equivalent to rJν0 (r ) − βνJν (r ) = 0, rJν0 (r ) + (1 − β − ν)Jν (r ) = 0 and rJν0 (r ) + (2 − 2β − ν)Jν (r ) = 0, respectively.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

14 / 35

Proof I Now, we prove the statement of part a when ν ∈ (−1, 0). First observe that the counterpart of (7), that is,   −|z| z Re ≥ α−z α + |z|

(9)

is valid for all p α ∈ R and z ∈ C such that α > |z|. Indeed, if we have z = x + iy and m = |z| = x 2 + y 2 , then (9) is equivalent to α(α − m)(m + x) ≥ 0, which is clearly true. By using (9), we obtain for all ν > −1, n ∈ {1, 2, . . . } and z ∈ D(0, jν,1 ) the inequality   −|z|2 z2 Re 2 ≥ 2 , (10) 2 jν,n − z jν,n + |z|2 which in turn implies that    X 2z 2 X 2|z|2 i|z|fν0 (i|z|) zfν0 (z) 1 ≥1+ 1 Re = 1 − Re  = . fν (z) ν ν fν (i|z|) j2 − z2 j 2 + |z|2 n≥1 ν,n n≥1 ν,n 

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

15 / 35

Proof

I This time we have equality if z = i|z| = ir , and from the above inequality we conclude that the first inequality in (4) holds if and only if |z| < xν,β , where xν,β denotes the positive root of the equation irfν0 (ir )/fν (ir ) = β, which is equivalent to rIν0 (r ) − βνIν (r ) = 0. All we need to prove is that xν,β is unique and xν,β < jν,1 for all β ∈ [0, 1) and ν ∈ (−1, 0), since in order to use (10) we tacitly assumed that for all β ∈ [0, 1) and n ∈ {2, 3, . . .} we have D(0, xν,β ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν ∈ (−1, 0).

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

16 / 35

Proof

I This time we have equality if z = i|z| = ir , and from the above inequality we conclude that the first inequality in (4) holds if and only if |z| < xν,β , where xν,β denotes the positive root of the equation irfν0 (ir )/fν (ir ) = β, which is equivalent to rIν0 (r ) − βνIν (r ) = 0. All we need to prove is that xν,β is unique and xν,β < jν,1 for all β ∈ [0, 1) and ν ∈ (−1, 0), since in order to use (10) we tacitly assumed that for all β ∈ [0, 1) and n ∈ {2, 3, . . .} we have D(0, xν,β ) ⊂ D(0, jν,1 ) ⊂ D(0, jν,n ) when ν ∈ (−1, 0). I For this recall that in case −1 < ν < −α the Dini function z 7→ zJν0 (z) + αJν (z) has all its zeros real and a single pair of conjugate purely imaginary zeros [Wa, p. 597]. Moreover, due to Ismail and Muldoon [IM, eq. (3.2)] we know that if ±iξ (ξ real) denote the purely imaginary zeros of the Dini function z 7→ zJν0 (z) + αJν (z), then ξ2 < −

Árpád Baricz (Babe¸s-Bolyai University)

α+ν 2 jν,1 . 2+α+ν

The radius of convexity of Bessel functions

October 09, 2013

16 / 35

Proof

I This in turn implies that 2 xν,β <−

ν(1 − β) 2 2 jν,1 < jν,1 , 2 + ν(1 − β)

as we required. Finally, consider the function qν : (0, ∞) → R, defined by qν (r ) = rIν0 (r )/Iν (r ) − βν. By using the asymptotic relations for small and large values of r for the function r 7→ Iν (r ), it can be verified that rIν0 (r )/Iν (r ) tends to ν as r → 0, and tends to infinity as r → ∞. Moreover, it is known (see for example [Ba]) that the function r 7→ rIν0 (r )/Iν (r ) is increasing on (0, ∞) for all ν > −1. Thus the function qν is increasing, qν (r ) tends to ν(1 − β) < 0 as r → 0, and tends to infinity as r → ∞. Consequently, the graph of qν intersects the r -axis only once, and thus the equation rIν0 (r ) − βνIν (r ) = 0 has only one solution. This completes the proof.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

17 / 35

The radius of convexity of normalized Bessel functions Theorem a. If ν > 0 and α ∈ [0, 1), then the radius of convexity of order α of the function fν is the smallest positive root of the equation   0 rJ 00 (r ) rJν (r ) 1 1 + 0ν + −1 = α. Jν (r ) ν Jν (r ) b. If ν > −1 and α ∈ [0, 1), then the radius of convexity of order α of the function gν is the smallest positive root of the equation 1+r

rJν+2 (r ) − 3Jν+1 (r ) = α. Jν (r ) − rJν+1 (r )

c. If ν > −1 and α ∈ [0, 1), then the radius of convexity of order α of the function hν is the smallest positive root of the equation 1

1

1+r2

Árpád Baricz (Babe¸s-Bolyai University)

1

1

r 2 Jν+2 (r 2 ) − 4Jν+1 (r 2 ) 1

1

1

= α.

2Jν (r 2 ) − r 2 Jν+1 (r 2 )

The radius of convexity of Bessel functions

October 09, 2013

18 / 35

The radius of convexity of normalized Bessel functions Theorem a. If ν > 0 and α ∈ [0, 1), then the radius of convexity of order α of the function fν is the smallest positive root of the equation   0 rJ 00 (r ) rJν (r ) 1 1 + 0ν + −1 = α. Jν (r ) ν Jν (r ) b. If ν > −1 and α ∈ [0, 1), then the radius of convexity of order α of the function gν is the smallest positive root of the equation 1+r

rJν+2 (r ) − 3Jν+1 (r ) = α. Jν (r ) − rJν+1 (r )

c. If ν > −1 and α ∈ [0, 1), then the radius of convexity of order α of the function hν is the smallest positive root of the equation 1

1

1+r2

1

1

r 2 Jν+2 (r 2 ) − 4Jν+1 (r 2 ) 1

1

1

= α.

2Jν (r 2 ) − r 2 Jν+1 (r 2 )

0 Moreover, rαc (fν ) < jν,1 < jν,1 , rαc (gν ) < αν,1 < jν,1 , rαc (hν ) < βν,1 < jν,1 , where jν,1 and 0 jν,1 are the first positive zeros of Jν and Jν0 , αν,1 is the first positive zero of z 7→ (1 − ν)Jν (z) + zJν0 (z), βν,1 is the first positive zero of z 7→ (2 − ν)Jν (z) + zJν0 (z). Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

18 / 35

Proof of part a I Observe that 1+

zfν00 (z) zJ 00 (z) = 1 + 0ν + 0 fν (z) Jν (z)



 0 zJν (z) 1 −1 . ν Jν (z)

Now, recall the following infinite product representations [OLBC, p. 235]
ν
ν−1   1 1 Y Y z z z2 z2 2 1− 2 , Jν0 (z) = 2 1 − 02 , Jν (z) = Γ(ν + 1) 2Γ(ν) jν,n jν,n n≥1 n≥1 0 where jν,n and jν,n are the nth positive roots of Jν and Jν0 , respectively.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

19 / 35

Proof of part a I Observe that 1+

zfν00 (z) zJ 00 (z) = 1 + 0ν + 0 fν (z) Jν (z)



 0 zJν (z) 1 −1 . ν Jν (z)

Now, recall the following infinite product representations [OLBC, p. 235]
ν
ν−1   1 1 Y Y z z z2 z2 2 1− 2 , Jν0 (z) = 2 1 − 02 , Jν (z) = Γ(ν + 1) 2Γ(ν) jν,n jν,n n≥1 n≥1 0 where jν,n and jν,n are the nth positive roots of Jν and Jν0 , respectively.Logarithmic differentiation yields

X 2z 2 X 2z 2 zJν0 (z) zJ 00 (z) =ν− , 1 + 0ν =ν− , 2 2 Jν (z) Jν (z) j −z j 02 − z 2 n≥1 ν,n n≥1 ν,n which implies that zf 00 (z) 1 + 0ν =1− fν (z)

Árpád Baricz (Babe¸s-Bolyai University)



X X 2z 2 2z 2 1 −1 − . 2 02 ν j − z 2 n≥1 jν,n − z2 n≥1 ν,n

The radius of convexity of Bessel functions

October 09, 2013

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Proof of part a I Now, suppose that ν ∈ (0, 1]. By using the inequality   |z| z , ≥ Re µ − |z| µ−z 0 where z ∈ C and µ ∈ R such that µ > |z|, then for all z ∈ D(0, jν,1 ) we obtain the inequality    X X 2r 2 zf 00 (z) 2r 2 1 Re 1 + 0ν − , ≥1− −1 02 fν (z) ν j 2 − r 2 n≥1 jν,n − r2 n≥1 ν,n

(11)

where |z| = r .

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

20 / 35

Proof of part a I Now, suppose that ν ∈ (0, 1]. By using the inequality   |z| z , ≥ Re µ − |z| µ−z 0 where z ∈ C and µ ∈ R such that µ > |z|, then for all z ∈ D(0, jν,1 ) we obtain the inequality    X X 2r 2 zf 00 (z) 2r 2 1 Re 1 + 0ν − , ≥1− −1 02 fν (z) ν j 2 − r 2 n≥1 jν,n − r2 n≥1 ν,n

(11)

where |z| = r .Moreover, observe that if we use the inequality     |z| |z| z z λ Re − Re ≥λ − , a−z b−z a − |z| b − |z| where a > b > 0, z ∈ C and λ ∈ [0, 1], |z| < b, then we get that (11) is also valid when ν > 1.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

20 / 35

Proof of part a I Now, suppose that ν ∈ (0, 1]. By using the inequality   |z| z , ≥ Re µ − |z| µ−z 0 where z ∈ C and µ ∈ R such that µ > |z|, then for all z ∈ D(0, jν,1 ) we obtain the inequality    X X 2r 2 zf 00 (z) 2r 2 1 Re 1 + 0ν − , ≥1− −1 02 fν (z) ν j 2 − r 2 n≥1 jν,n − r2 n≥1 ν,n

(11)

where |z| = r .Moreover, observe that if we use the inequality     |z| |z| z z λ Re − Re ≥λ − , a−z b−z a − |z| b − |z| where a > b > 0, z ∈ C and λ ∈ [0, 1], |z| < b, then we get that (11) is also valid when 0 interlace according to the inequalities ν > 1.Here we used that the zeros jν,n and jν,n [OLBC, p. 235] 0 0 0 ν ≤ jν,1 < jν,1 < jν,2 < jν,2 < jν,3 < . . .. (12) 0 Now, the above deduced inequality implies for r ∈ (0, jν,1 )    00 rf 00 (r ) zf (z) inf Re 1 + 0ν = 1 + ν0 . z∈D(0,r ) fν (z) fν (r ) Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

20 / 35

Proof of part a 0 I On the other hand, the function uν : (0, jν,1 ) → R, defined by

uν (r ) = 1 +

rfν00 (r ) , fν0 (r )

is strictly decreasing since uν0 (r ) = − <



X n≥1

X 2 02 X 4rjν,n 4rjν,n 1 − −1 2 02 2 2 ν (j − r ) (j − r 2 )2 n≥1 ν,n n≥1 ν,n 2 02 X 4rjν,n 4rjν,n − <0 2 02 (jν,n − r 2 )2 n≥1 (jν,n − r 2 )2

0 for ν > 0 and r ∈ (0, jν,1 ).

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

21 / 35

Proof of part a 0 I On the other hand, the function uν : (0, jν,1 ) → R, defined by

uν (r ) = 1 +

rfν00 (r ) , fν0 (r )

is strictly decreasing since uν0 (r ) = − <



X n≥1

X 2 02 X 4rjν,n 4rjν,n 1 − −1 2 02 2 2 ν (j − r ) (j − r 2 )2 n≥1 ν,n n≥1 ν,n 2 02 X 4rjν,n 4rjν,n − <0 2 02 (jν,n − r 2 )2 n≥1 (jν,n − r 2 )2

0 0 for ν > 0 and r ∈ (0, jν,1 ).Here we used again that the zeros jν,n and jν,n interlace and q 0 for all n ∈ {1, 2, . . . }, ν > 0 and r < jν,1 jν,1 we have that 2 02 02 2 jν,n (jν,n − r 2 )2 < jν,n (jν,n − r 2 )2 . 0 Observe also that limr &0 uν (r ) = 1 > α and limr %jν,1 uν (r ) = −∞, which means that   00 zf (z) > α, if and only if r1 is the unique root of for z ∈ D(0, r1 ) we have Re 1 + 0ν fν (z)

1+

rfν00 (r ) = α, fν0 (r )

0 situated in (0, jν,1 ). Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

21 / 35

Theorem The function fν is convex of order α ∈ (0, 1) in D if and only if ν ≥ να (fν ), where να (fν ) is the unique root of the equation ν(ν 2 − 1)Jν2 (1) + (1 − ν)(Jν0 (1))2 = ανJν (1)Jν0 (1), situated in (ν ∗ , ∞), where ν ∗ ' 0.3901 . . . is the root of the equation Jν0 (1) = 0. Moreover, fν is convex in D if and only if ν ≥ 1.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

22 / 35

Proof

I According to (11) for z ∈ D we obtain that    X X 2r 2 zf 00 (z) 1 2r 2 ≥1− Re 1 + 0ν − −1 2 2 fν (z) ν j −r j 02 − r 2 n≥1 ν,n n≥1 ν,n  X X 2 1 2 ≥1− − −1 2 02 ν j − 1 n≥1 jν,n − 1 n≥1 ν,n   0 00 J (1) Jν (1) f 00 (1) 1 = 1 + ν0 + −1 = 1 + ν0 . Jν (1) ν Jν (1) fν (1)

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

23 / 35

Proof

I According to (11) for z ∈ D we obtain that    X X 2r 2 zf 00 (z) 1 2r 2 ≥1− Re 1 + 0ν − −1 2 2 fν (z) ν j −r j 02 − r 2 n≥1 ν,n n≥1 ν,n  X X 2 1 2 ≥1− − −1 2 02 ν j − 1 n≥1 jν,n − 1 n≥1 ν,n   0 00 J (1) Jν (1) f 00 (1) 1 = 1 + ν0 + −1 = 1 + ν0 . Jν (1) ν Jν (1) fν (1) Now, consider the function u : (ν ∗ , ∞) → R, defined by   0  X X Jν00 (1) Jν (1) 1 1 2 2 u(ν) = 1 + 0 + −1 =1− −1 − . 2 02 Jν (1) ν Jν (1) ν j − 1 j −1 n≥1 ν,n n≥1 ν,n We note that this function is well defined since Jν (1) > 0 and Jν0 (1) > 0 when ν > ν ∗ .

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

23 / 35

Proof I By using 2


x ν 2

Jν (x) = √ πΓ ν + for ν >

− 21

Z
1 2

1

1

(1 − t 2 )ν− 2 cos(xt)dt

0

we get
ν Z 1 2 12 1
Jν (1) = √ (1 − t 2 )ν− 2 cos(t)dt > 0. πΓ ν + 12 0

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

(13)

October 09, 2013

24 / 35

Proof I By using 2


x ν 2

Jν (x) = √ πΓ ν + for ν >

− 21

Z
1 2

1

1

(1 − t 2 )ν− 2 cos(xt)dt

0

we get
ν Z 1 2 12 1
Jν (1) = √ (1 − t 2 )ν− 2 cos(t)dt > 0. πΓ ν + 12 0

(13)

I Moreover, since ν 7→ jν,n is strictly increasing on (0, ∞) for each n ∈ {1, 2, . . . } (see [OLBC, p. 236]), it follows that    0  X X 4jν,n djν,n Jν (1) d d  2  dν ν− = = 1 + >0 dν Jν (1) dν j2 − 1 (j 2 − 1)2 n≥1 ν,n n≥1 ν,n for ν > 0. This means that if ν > ν ∗ , then Jν0 (1)/Jν (1) > Jν0 ∗ (1)/Jν ∗ (1) = 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

24 / 35

Proof I By using 2


x ν 2

Jν (x) = √ πΓ ν + for ν >

− 21

Z
1 2

1

1

(1 − t 2 )ν− 2 cos(xt)dt

0

we get
ν Z 1 2 12 1
Jν (1) = √ (1 − t 2 )ν− 2 cos(t)dt > 0. πΓ ν + 12 0

(13)

I Moreover, since ν 7→ jν,n is strictly increasing on (0, ∞) for each n ∈ {1, 2, . . . } (see [OLBC, p. 236]), it follows that    0  X X 4jν,n djν,n Jν (1) d d  2  dν ν− = = 1 + >0 dν Jν (1) dν j2 − 1 (j 2 − 1)2 n≥1 ν,n n≥1 ν,n for ν > 0. This means that if ν > ν ∗ , then Jν0 (1)/Jν (1) > Jν0 ∗ (1)/Jν ∗ (1) = 0. I Now, in what follows we show that u is strictly increasing. For this we distinguish two 0 cases. First we consider that ν ∈ (ν ∗ , 1]. Since the functions ν 7→ jν,n and ν 7→ jν,n are strictly increasing on [0, ∞) for each n ∈ {1, 2, . . . } (see [OLBC, p. 236]), it follows that 2 02 the functions ν 7→ 2/(jν,n − 1) and ν 7→ 2/(jν,n − 1) are strictly decreasing on [0, ∞) for each n ∈ {1, 2, . . . }, and consequently u is strictly increasing on (ν ∗ , 1]. Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

24 / 35

Proof I Suppose that ν > 1. In this case we have 0

u 0 (ν) =

  djν,n 0 djν,n X jν,n 1 X 1 X jν,n dν 2 dν − 4 1 − + 4 . 2 2 02 2 ν2 ν j − 1 (j − 1) (j − 1)2 ν,n ν,n ν,n n≥1 n≥1 n≥1

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

25 / 35

Proof I Suppose that ν > 1. In this case we have 0

u 0 (ν) =

  djν,n 0 djν,n X jν,n 1 X 1 X jν,n dν 2 dν − 4 1 − + 4 . 2 2 02 2 ν2 ν j − 1 (j − 1) (j − 1)2 ν,n ν,n ν,n n≥1 n≥1 n≥1

0 Recall that for any n ∈ {1, 2, . . . } the derivative of jν,n and jν,n with respect to ν can be written as [OLBC, p. 236] Z ∞ djν,n = 2jν,n K0 (2jν,n sinh(t))e−2νt dt, (14) dν 0 Z ∞ 0 0 2jν,n djν,n 02 0 = 02 (jν,n cosh(2t) − ν 2 )K0 (2jν,n sinh(t))e−2νt dt, dν jν,n − ν 2 0

where K0 stands for the modified Bessel function of the second kind and zero order (see [OLBC, p. 252]).

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

25 / 35

Proof I Suppose that ν > 1. In this case we have 0

u 0 (ν) =

  djν,n 0 djν,n X jν,n 1 X 1 X jν,n dν 2 dν − 4 1 − + 4 . 2 2 02 2 ν2 ν j − 1 (j − 1) (j − 1)2 ν,n ν,n ν,n n≥1 n≥1 n≥1

0 Recall that for any n ∈ {1, 2, . . . } the derivative of jν,n and jν,n with respect to ν can be written as [OLBC, p. 236] Z ∞ djν,n = 2jν,n K0 (2jν,n sinh(t))e−2νt dt, (14) dν 0 Z ∞ 0 0 2jν,n djν,n 02 0 = 02 (jν,n cosh(2t) − ν 2 )K0 (2jν,n sinh(t))e−2νt dt, dν jν,n − ν 2 0

where K0 stands for the modified Bessel function of the second kind and zero order (see [OLBC, p. 252]).Observe that for ν > 1, n ∈ {1, 2, . . . } and t > 0 we have 02 jν,n cosh(2t) − ν 2 1 >1>1− , 02 ν jν,n − ν2

(15)

02 2 jν,n jν,n > , 02 2 (jν,n − 1)2 (jν,n − 1)2

(16)

2 02 2 02 2 02 2 02 since according to (12) we have that jν,n > jν,n and jν,n jν,n > jν,1 jν,1 > j1,1 j1,1 > 1. Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

25 / 35

Proof I On the other hand, we know that K0 is strictly decreasing on (0, ∞), and this implies that for each ν > 1, n ∈ {1, 2, . . . } and t > 0 we have 0 K0 (2jν,n sinh(t)) > K0 (2jν,n sinh(t)).

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

26 / 35

Proof I On the other hand, we know that K0 is strictly decreasing on (0, ∞), and this implies that for each ν > 1, n ∈ {1, 2, . . . } and t > 0 we have 0 K0 (2jν,n sinh(t)) > K0 (2jν,n sinh(t)).

Combining this with (15) and (16) we obtain 0

0 djν,n 02 02 X Z ∞ jν,n X jν,n cosh(2t) − ν 2 jν,n 0 dν = 8 K (2jν,n sinh(t))e−2νt dt 4 02 02 02 2 2 2 0 (j − 1) j − ν (j − 1) ν,n ν,n n≥1 0 n≥1 ν,n 02 2 02 X Z ∞ jν,n cosh(2t) − ν jν,n >8 K (2jν,n sinh(t))e−2νt dt 02 02 2 2 0 j − ν (j − 1) ν,n ν,n 0 n≥1  2 XZ ∞  jν,n 1 >8 1− K0 (2jν,n sinh(t))e−2νt dt 2 ν (jν,n − 1)2 0 n≥1

  djν,n 1 X jν,n dν =4 1− , ν (j 2 − 1)2 n≥1 ν,n which implies that u 0 (ν) > 0 for ν > 1, and thus the function u is strictly increasing on (1, ∞), and hence on the whole (ν ∗ , ∞). Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

26 / 35

Proof I Consequently, if ν ≥ να = να (fν ), then we get the inequality u(ν) ≥ u(να ). This in turn implies that να is the smallest value having the property that the condition ν ≥ να implies that for all z ∈ D we have   f 00 (z) Re 1 + z ν0 > u(να ) = α. fν (z)

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

27 / 35

Proof I Consequently, if ν ≥ να = να (fν ), then we get the inequality u(ν) ≥ u(να ). This in turn implies that να is the smallest value having the property that the condition ν ≥ να implies that for all z ∈ D we have   f 00 (z) Re 1 + z ν0 > u(να ) = α. fν (z) Thus, we proved that the function fν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (fν ), where να = να (fν ) is the unique root of the equation u(ν) = α, that is,   0 J 00 (1) Jν (1) 1 1 + ν0 + −1 = α. Jν (1) ν Jν (1)

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

27 / 35

Proof I Consequently, if ν ≥ να = να (fν ), then we get the inequality u(ν) ≥ u(να ). This in turn implies that να is the smallest value having the property that the condition ν ≥ να implies that for all z ∈ D we have   f 00 (z) Re 1 + z ν0 > u(να ) = α. fν (z) Thus, we proved that the function fν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (fν ), where να = να (fν ) is the unique root of the equation u(ν) = α, that is,   0 J 00 (1) Jν (1) 1 1 + ν0 + −1 = α. Jν (1) ν Jν (1) Since Jν is a particular solution of the Bessel differential equation (see [OLBC, p. 217]), it follows that Jν00 (1) + Jν0 (1) + (1 − ν 2 )Jν (1) = 0, and by using this the above equation can be rewritten as ν(ν 2 − 1)Jν2 (1) + (1 − ν)(Jν0 (1))2 = ανJν (1)Jν0 (1).

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

27 / 35

Proof I Consequently, if ν ≥ να = να (fν ), then we get the inequality u(ν) ≥ u(να ). This in turn implies that να is the smallest value having the property that the condition ν ≥ να implies that for all z ∈ D we have   f 00 (z) Re 1 + z ν0 > u(να ) = α. fν (z) Thus, we proved that the function fν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (fν ), where να = να (fν ) is the unique root of the equation u(ν) = α, that is,   0 J 00 (1) Jν (1) 1 1 + ν0 + −1 = α. Jν (1) ν Jν (1) Since Jν is a particular solution of the Bessel differential equation (see [OLBC, p. 217]), it follows that Jν00 (1) + Jν0 (1) + (1 − ν 2 )Jν (1) = 0, and by using this the above equation can be rewritten as ν(ν 2 − 1)Jν2 (1) + (1 − ν)(Jν0 (1))2 = ανJν (1)Jν0 (1). I In particular, when α = 0 the above equation becomes (ν − 1)(ν(ν + 1)Jν2 (1) − (Jν0 (1))2 ) = 0, and by using the relation [OLBC, p. 222] zJν0 (z) = zJν−1 (z) − νJν (z),

(17)

we obtain that the above equation can be rewritten as 2 (ν − 1)(νJν2 (1) + 2νJν−1 (1)Jν (1) − Jν−1 (1)) = 0.

This equation has two roots: ν1 = 1 and ν2 ' 0.1246. . ., but only ν1 is situated in (ν ∗ , ∞). Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

27 / 35

Theorem The function gν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (gν ), where να (gν ) is the unique root of the equation (2ν + α − 2)Jν+1 (1) = αJν (1), situated in [0, ∞). In particular, gν is convex in D if and only if ν ≥ 1.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

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Theorem The function gν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (gν ), where να (gν ) is the unique root of the equation (2ν + α − 2)Jν+1 (1) = αJν (1), situated in [0, ∞). In particular, gν is convex in D if and only if ν ≥ 1.

Theorem The function hν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (hν ), where να (hν ) is the unique root of the equation (2ν + α − 3)Jν+1 (1) = (2α − 1)Jν (1), situated in [0, ∞). In particular, hν is convex if and only if ν ≥ ν0 (hν ), where ν0 (hν ) ' 0.306 . . . is the unique root of the equation (2ν − 3)Jν+1 (1) + Jν (1) = 0. Moreover, in particular, the function hν is convex of order

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

1 2

if and only if ν ≥ 32 .

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I We note that the convex functions does not need to be normalized. In other words, the analytic and univalent function f : D → C satisfying f 0 (0) 6= 0 is said to be convex of order α ∈ [0, 1) if and only if   f 00 (z) Re 1 + z 0 >α f (z) for all z ∈ D.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

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I We note that the convex functions does not need to be normalized. In other words, the analytic and univalent function f : D → C satisfying f 0 (0) 6= 0 is said to be convex of order α ∈ [0, 1) if and only if   f 00 (z) Re 1 + z 0 >α f (z) for all z ∈ D. I In 1995 Selinger [Se] by using the method of differential subordinations proved that the function ϕν : D → C, defined by ϕν (z) =

√ ν hν (z) 1 = 2ν Γ(ν + 1)z − 2 Jν ( z) = 1 − z + ..., z 4(ν + 1)

is convex if ν ≥ − 14 . In 2009 Szász and Kupán [SK], by using a completely different approach, improved this result, and proved that ϕν is convex in D if ν ≥ ν1 ' −1.4069 . . . , where ν1 is the root of the equation 4ν 2 + 17ν + 16 = 0. Recently, Baricz and Ponnusamy [BP] presented four improvements of the above result, and their best result was the following [BP, Theorem 3]: the function ϕν is convex in D if ν ≥ ν2 ' −1.4373 . . . , where ν2 is the unique root of the equation 2ν Γ(ν + 1)(Iν+2 (1) + 2Iν+1 (1)) = 2. Moreover, Baricz and Ponnusamy [BP] conjectured that ϕν is convex in D if and only if ν ≥ −1.875. Now, we are able to disprove this conjecture and to find the radius of convexity of the function ϕν . Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

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Theorem If ν > −2 and α ∈ [0, 1), then the radius of convexity of order α of the function ϕν is the smallest positive root of the equation 1

1

r 2 Jν (r 2 ) 1

− ν = α.

2Jν+1 (r 2 ) Moreover, we have rαc (ϕν ) < jν+1,1 .

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

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Theorem If ν > −2 and α ∈ [0, 1), then the radius of convexity of order α of the function ϕν is the smallest positive root of the equation 1

1

r 2 Jν (r 2 ) 1

− ν = α.

2Jν+1 (r 2 ) Moreover, we have rαc (ϕν ) < jν+1,1 .

Theorem The function ϕν is convex of order α ∈ [0, 1) in D if and only if ν ≥ να (ϕν ), where να (ϕν ) is the unique root of the equation (2ν + 2α)Jν+1 (1) = Jν (1), situated in (ν ? , ∞) , where ν ? ' −1.7744 . . . is the root of the equation Jν+1 (1) = 0. In particular, ϕν is convex in D if and only if ν ≥ ν0 (ϕν ), where ν0 (ϕν ) ' −1.5623 . . . is the unique root of the equation Jν (1) = 2νJν+1 (1), situated in (ν ? , ∞) .

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

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Preliminary results For more details see [Wa, p. 482] and [Wa, p. 198].

Lemma If ν > −1 and a, b ∈ R, then z 7→ aJν (z) + bzJν0 (z) has all its zeros real, except the case when a/b + ν < 0. In this case it has two purely imaginary zeros beside the real roots. Moreover, if ν > −1 and a, b ∈ R such that a2 + b2 6= 0, then no function of the type z 7→ aJν (z) + bzJν0 (z) can have a repeated zero other than z = 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

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Preliminary results For more details see [Wa, p. 482] and [Wa, p. 198].

Lemma If ν > −1 and a, b ∈ R, then z 7→ aJν (z) + bzJν0 (z) has all its zeros real, except the case when a/b + ν < 0. In this case it has two purely imaginary zeros beside the real roots. Moreover, if ν > −1 and a, b ∈ R such that a2 + b2 6= 0, then no function of the type z 7→ aJν (z) + bzJν0 (z) can have a repeated zero other than z = 0.

Lemma (1)

(2)

If ν > −1, then for the Bessel functions of the third kind Hν and Hν the following asymptotic expansions are valid Hν(1) (w) =

Hν(2) (w) =





2 πw

2 πw

1 2

1 2

1

1

ei(w− 2 νπ− 4 π) (1 + η1,ν (w)), 1

1

e−i(w− 2 νπ− 4 π) (1 + η2,ν (w)),

where η1,ν (w) and η2,ν (w) are O(1/w) when |w| is large.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

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Preliminary results

Lemma Let αν,n be the n-th positive root of the equation Jν (z) − zJν+1 (z) = 0. If ν > −1, then the following development holds X zJν+2 (z) − 3Jν+1 (z) gν00 (z) 2z = =− . 2 0 gν (z) Jν (z) − zJν+1 (z) α − z2 ν,n n≥1

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

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Preliminary results

Lemma Let αν,n be the n-th positive root of the equation Jν (z) − zJν+1 (z) = 0. If ν > −1, then the following development holds X zJν+2 (z) − 3Jν+1 (z) gν00 (z) 2z = =− . 2 0 gν (z) Jν (z) − zJν+1 (z) α − z2 ν,n n≥1

Lemma Let βν,n denote the nth positive root of (2 − ν)Jν (z) + zJν0 (z) = 0. If ν > −1, then the following development holds 1

1

1

1

ν(ν − 2)Jν (z 2 ) + (3 − 2ν)z 2 Jν0 (z 2 ) + zJν00 (z 2 ) 1 2

2(2 − ν)Jν (z ) + 2z

Árpád Baricz (Babe¸s-Bolyai University)

1 2

1

Jν0 (z 2 )

The radius of convexity of Bessel functions

=−

X n≥1

2z . 2 βν,n −z

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Preliminary results

Lemma If ν ≥ 0, then αν,1 > 1, where αν,1 denotes the first positive root of the equation Jν (x) − xJν+1 (x) = 0. Similarly, if ν ≥ 0, then βν,1 > 1, where βν,1 denotes the first positive root of the equation 2Jν (x) − xJν+1 (x) = 0.

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

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Preliminary results

Lemma If ν ≥ 0, then αν,1 > 1, where αν,1 denotes the first positive root of the equation Jν (x) − xJν+1 (x) = 0. Similarly, if ν ≥ 0, then βν,1 > 1, where βν,1 denotes the first positive root of the equation 2Jν (x) − xJν+1 (x) = 0.

Lemma For ν > −1 let γν,n be the nth positive root of the equation γJν (z) + zJν0 (z) = 0. If ν + γ ≥ 0, then the function ν 7→ γν,n is strictly increasing on (−1, ∞) for n ∈ {1, 2, . . . } fixed. For the last result see [La, p. 196].

Árpád Baricz (Babe¸s-Bolyai University)

The radius of convexity of Bessel functions

October 09, 2013

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References [Ba] Á. Baricz: Turán type inequalities for modified Bessel functions, Bull. Aust. Math. Soc. 82 (2010) 254–264. [BKS] Á. Baricz, P.A. Kupán, R. Szász: The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc. (in press), arXiv:1202.1504. [BP] Á. Baricz, S. Ponnusamy: Starlikeness and convexity of generalized Bessel functions, Integr. Transforms Spec. Funct. 21 (2010) 641–653. [BS] Á. Baricz, R. Szász: The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl. (submitted), arXiv:1302.4222. [Br1] R.K. Brown: Univalence of Bessel functions, Proc. Amer. Math. Soc. 11(2) (1960) 278–283. [Br2] R.K. Brown: Univalent solutions of W 00 + pW = 0, Canad. J. Math. 14 (1962) 69–78. [HM] T.L. Hayden, E.P. Merkes: Chain sequences and univalence, Illinois J. Math. 8 (1964) 523–528. [IM] M.E.H. Ismail, M.E. Muldoon: Bounds for the small real and purely imaginary zeros of Bessel and related functions, Methods Appl. Anal. 2(1) (1995) 1–21. [KT] E. Kreyszig, J. Todd: The radius of univalence of Bessel functions, Illinois J. Math. 4 (1960) 143–149. [La] L.J. Landau: Ratios of Bessel functions and roots of αJν (x) + xJν0 (x) = 0, J. Math. Anal. Appl. 240 (1999) 174–204. [Ne] Z. Nehari: The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949) 545–551.

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The radius of convexity of Bessel functions

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References

[Ro] M.S. Robertson: Schlicht solutions of W 00 + pW = 0, Trans. Amer. Math. Soc. 76 (1954) 254–274. [Se] V. Selinger: Geometric properties of normalized Bessel functions, Pure Math. Appl. 6 (1995) 273–277. [Sz] R. Szász: On starlikeness of Bessel functions of the first kind, In: Proceedings of the 8th Joint Conference on Mathematics and Computer Science, Komárno, Slovakia, 2010, 9pp. [SK] R. Szász, P.A. Kupán: About the univalence of the Bessel functions, Stud. Univ. Babe¸s-Bolyai Math. 54(1) (2009) 127–132. [Ba] Á. Baricz: Generalized Bessel Functions of the First Kind, Springer, vol. 1994, Berlin, 2010. [OLBC] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.): NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010. [Wa] G.N. Watson: A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944.

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