Far Eost J. Math. Sci. (FJMS) l2(2) (2004),191-201

ON SUFFICIENT CONDITIONS FOR CARATHEODORYFUNCTIONS V. RAVICHANDRAN

ANd M. JAYAMALA

( ReceivedJuly 31, 2003 )

Submitted by SaburouSaitoh

Abstract Recently Nunokawa et al. flndian J. Pure Appl. Math. 33(9) (2002), 1385-13901have obtained some sufficient condition for analytic function defined on I z | < t to have positive real part. In this paper, we find sufficient condition for the subordination p(z) < q(z) to hold. Some applicationsof these results are also discu.ssed.

1. Introduction Let As be the classof all functions f (") = z * a2z2 * agzs + "' which are analytic in L={z;lrl.t}. p(z) =l+

Let A be the class of all functions

pyz + p2z2 +... which are analytic in A. The class P of

Caratheodory functions consists of functions p(z) e A having positive real part. Recently Nunokawa et al. [2] have found out conditions on o, p and w(z\ such that each of the following differential subordination implies p(z) e P: . ap(z)z + PzP'(z)< w(z), 2000 Mathematics Subject Classi{ication:30C45. Key words and phrases: Caratheodory function, differential subordination. @ 2004 Pushpa Publishing House

L92

V. RAVICHANDRAN and M. JAYAMALA

. ap(z)+ pz1@\)< w(z), p\z )

. op(")+ g zet' ?) <,,u(z). plz)-

Note that the conclusion p(z) e P can be written as p(z) < (L + z)l$ - z). In this paper, we find sufficient conditions for the subordination p(z) < q(z) to hold. Our results include the results obtained by Nunokawa et al. [2]. We also give some application of our results to obtain sufficient conditions for starlikeness. We need the following result of Miller and Mocanu [1] to prove our main result: Theorem A. Let q(z) be wiualent irt th,eudt disk A^an d 0 an d g be analytic hr a domain D containirlg q(L) taith S(u) + 0 when w e q(t). Set Q@) = zq'(z)Q@GD,h(z) = e(q(")) + QQ). Supposethat t. QG) is starlike uiualent itt A,,and

2.Re49>oforzea,. Q@)

rf e(p(e))+ zp'(z)g(p(")) < e(q(r)) + zq'(z)g(q(")), then.p(z) < q(z) and q(z) is the bestdominant. 2. Sufficient Conditions We begin with the following: Theorem 1. Let a, p be any complexnumbers, p * 0. Let q(z) e A be ur,iualent irt A,and satisfy the followirtg conditiortsfor z e L: (9n

.l

(i) Rel # q(") + (zq'(z)lq'Q)+ 1)f > 0; t p ) (rl) zq'(z) is starlihe.

CARATHEODORYFUNCTIONS

193

Ifp(")eAsatisfies

+ Bzp'(z) . ort) + Bzq'(z), ap(z)z th.enp(z\ < q(z) arudq is the bestdomhtanfi. Proof. Let O(ru): s.w2 and Q(ur)= B. The" 0(,r) + 0 and 0(r), 0(r) are analytic in C. Let the functions @(e)and lt(z) be defined by QQ) = zq'(z)0(q(r)): Bzq'(z), h(z) = e(q(r)) + QQ) = aq(z)z+ Bzq'(z)Then we seethat QQ) is starlike and

o"#E=Re

=o"[? 'o qe)+(+&.')] The differential subordination ap(z)z+ pzp'(z) . o{@) + Pzq'(z) becomes e(p(r)) + zp'(z)$(p(")) < e(q(r)) + zq'(z)$(q(")'1. The result follows on an application of Theorem A. Let q(z) be the function

q(z)=*#,

-1
Then we have z q '''( - -\ - ( A - B ) "

(L* B"f

'

V. RAVICHANDRAN ANdM. JAYAMAI,A

t94

Let g(z) - zq'(z)-Then it follows that -

= | Bz TTBz' iW zq'Q\

with z = reio, we have o" rt'E

zg'r(zr): g(z)

I - B2r2z

>0.

| + Bzrz + 2Br cos o

have Therefor e zq'(z) is starlike in A' Also we

y +62 _ I+ Bz' The function where Y=1+ZalB and 6 = Z a A l p - 8 .

w(z) =

y+62 l+Bz

maps A into the disk

I

v-^B6 . l 6 - 7 { 1 .

I

r-8"

Iut-

l-Bz

Therefore

provided

Re(T-Bs)>ls-YBl or

Ru(r- 86)> l6 - fi| Therefore we have the following: -I< B < AsL' Let u' B satisfy R " ( y - B s ) > l s - T B l , Corollar y L. Let - ZaAlB - B' If p(z) e A a n d wherey = 1 + ZulB an'd' 6

CARATHEODORYFUNCTIONS

. "(*#)' +pzp'(z) op(")'

195

- 1< B < A < r,

*u t#,

1 + A z

thenp(z)< fu

By takin g A = I, B = -I, and cr and B to be real, then we have the following result of Nunokawa et al. [2]: Corollary

2. Let I + Zulp > 0- If p(z) e A art'd

(z). "(=)' + Bzp, up(z)z

.

;%,

then Re p(z) > 0. If q(z) is a convex function that maps A onto a region in the right half-plane, then the conditions (i) and (ii) of Theorem 1 are satisfied by q(z) whenever crB> 0. BY taking o < 6 < 1,

q(z):f=)', -

\ L

.

)

we have the following: Corollary 3. Let crB> 0. If p(z) e A arud -(1 + e)zs op(z)z+ Bzp'(z)<

" i ] r - , ).

286z /1 + z\6

o-E\t-")'

then I A's p@)l < 6n12. nurnbers,B * o. Let q(z) e A be Theorem 2. Let u, B beany complex bestarlikean'd uniualentirtL,,q(z\ * 0. Let QQ) = pzq'(z)lq@) (n

. '..l z e L" o"i;qG)+@Q'G)lQ(,))i'0,

Ifp(z)eAsatisfies

ap(z).e#< aq(z)*B+&, then, p(z) < q(z) and q is the best domhtan't'

196

V. RAVICHANDRAN and M. JAYAMALA

If q(z) is a convex function that maps A onto a region in the right half-plane and zq'(z)lqQ) is starlike, op > 0, then the conditions of Theorem 2 are satisfied. Proof. Let O(ru)= a"w and g(u) = gl*.Then 0@) + 0 and 0(r), $(ru) are analytic in a - {0} which contains q(A). Let the functions Q(z) and h(z) be defined by

e@)=zq'(z)Q@Q))=p+&, o zQ' (z) h (z ) = e(q(r)) + Q@ ) = aq(z)'- P -q6' Then we see that QQ) is starlike and azq'(z) + zQ'G) o^ zh'(z) _ - Ro rvv rvv QG) Pzq'(z)lq@)

=o.[fiqe)+8&],' The differential subordination "Pt@^) za' < aq(z)+ g'-qdQ) aP(z)* g p\. ) becomes e(p(')) + zp'(z)g(p(r)) < e(q(r)) + zq'(z)g(q(")). By an application of Theorem A, we get the result. Let q(z) = (t + Az)lQ + Bz). Then q(z) + O and maps A onto a circular region in the right half-plane. Let

(4

h ( z \ =q(z) '-\-/ 4 4 9- 3(r) = ,. .P)"- =. * Az)(r+ Bz)' We provethat the function h(z) is starlike.A computationshowsthat Az Bz H(") 't =zh'(z) 6 = r -- l' t . Ie . ' L * u " ) I

CARATHEODORY FUNCTIONS

197

:

=w'

I - ABz2

Now I-AB'2e2io ' ReH(eio)=Re ' (r+Areit)(t* Breio)

t + r ( A + B ) ( t - A B r 2 ) c o s o -A z B z r a l(r + Areit) (r * Bruie112 -.qBrz)(t + ABr2 + r(A + F)cos}) _ (t , o l(r + Areit) (r * Brris )12 p r o v i d e dI + A B r 2 + r ( A + B ) c o s g > 0 . s i n c e- 1 < B < A < I ,

l+ABr2

+ r(A + B)coso > (r - Ar)(r - Br)> 0 when (A + B) > 0. Also l+ ABr2 + r(A+ B)cos0 t (t * Ar)(1 + Br)> 0 when (A + B) < 0. Therefore we have the following: c o r o l l a r y 4 . L e t - 1 < B < A < L . L e t c r) 0 , p > 0 . I f p ( " ) e A a n ' d

o ( A- B ) z A ' ' a p ( z )'B. "p!\rz9) r.) o + t+Bz'PW'

thenp(z)<'i# BytakingA:|,8=_|,andcr=F=l,thenwehavethefollowing result of Nunokawa et al. [2]: Corollary

5. If P(z) e A attd

p( z )+ " R .tt!+ , P \z)

I \

then Rep(z) > o.

I

z"

198

V. RAVICHANDRAN and M. JAYAMALA

Theorem 3. Let c, p * 0 be any conplex numbers.Let q(z) e A be uriualent in A,,q(z) * O. Let QQ) = Bzq'(z)lq/)z be starlike and,

ze1,. o.{ffqe)z.ffi},0, Ifp(")eAsatisfies ' : \ - / '+' ' q (")'' . ap(z)* B"P'?) < -aq(z) Bza'@) p(")2

then p(z) < q(z) and q is tlte best dominantt. The proof of this theorem is similar to the proof of Theorem 2 and therefore omitted. As a special case, we have: Corollary

6. Let u and p be positiue numbers. If

-a' - q. a p ( -z )p 4 9 -. ' *o r - j ' - . ' F " . , -= *o1r -- j : * 9 ( t + "' z\t z) 2' | z (i+z)2 ;A then Re p(z) > 0. Proof.

The corollary follows from Theorem 3 by taking

q(z) =

(L + z)lQ - z) (after replacing F bv -p). Of course, the function Q@) =

- zpzle Norethat + z)2 is starlike.

. q&: fra(42

-ff (=)'

."991= 9cot29 = o. ' t + z ' with z = ei|,wehaveRe[-:p qQ)z *!-'. 2 L -'-' Qk)l P Note that Nunokawa et al. [2] have proved that: If

- p,49 *2": P(=\' ap(z) pl")z

2

\t

")

-2, 2'

then Re p(z) > 0. Our result gives the best dominant. By taking o = 0 in Theorem 3, we have the following:

199

FUNCTIONS CARATHEODORY in' A,,q(z) * 0- nrt Corollary 7. Let q(z)e A beurtiualen't

"19.

U'

q(z)"

starlike. If p(") e A satisfies

49 t "q'(")' p(")'

q(z)'

then. p(z) < q(z). The dominanrt q is the best domht'anfi.

3. Starlikeness Criteria In this section, we give application of our results for getting sufficient conditions for starlikeness. Note that the class S-[A, Bl consists of functions in ,4 for which Az. _1 < zf'(z) B < A
L e t a , p s a t i . t / yR e ( y - 8 6 ) > l 6 - - Y B l ,

where y = 1 + Zo,lpand' 6 = 2aAl9 - B' If f(") t As and

. uff#' . T&\'"(##)' #[t"-il#. u(, then f (") . S-[A, B]Proof. Let p(z) =

Then a computation showsthat #.

4 9 + p @ )1=++ 9 P@

I \z)

Therefore, we have

#[t"-il#.u('.+E)]

200

V. RAVICHANDRAN and M. JAYAMALA

=p(z) - il pe). r(#. [t"

"0,)]

= ap(z)z + Bzp'(z) r ++ABzz) 2**pn G (A-B)z <, ^" ([ 1 ) By an application of Corollary 1, we have the result. As a specialcase,we get the following. Corollary 8. Let | + zulB > 0. If f(") . As and,

. "(=)'. ;5, . T&)J #[t" -il#. u(' then /(z) e S-. By using Corollary 4 with p(z) = zf'(z)lf Q), *" have the following: T h e o r e m S . L e t- 1 < B < A < 1 . L e t o ) 0 , B > 0 . I f f ( " ) e A s a n d

-P#). u .(T' & )".I # *t f f i , (o then.f (") . S-[A, B]. In particular, we have Corollary 9. Let -1 < B < A
As and'

(A - B)z

then,f(") . S-[A, B]. '(z)lf Using Corollary 6 with p(z) = zf Q), *" have the following: Theorem 6. Let a anrdB bepositiue numbers.If f (") e Ao satisfies

20L

CARATHEODORY F'UNCTIONS

-p'-# "+& 1 , "f"(")

{

'9"

L + z

CX,;L - Z

^ -g,

(t + z)"

then,f(z) e 5.. By taking

0. we have the following

result of Obradovie and

Tuneski [3]. Corollary L0.If f (r) . As satisfies r' -' 7zf"(z) q<

-ZfT--

1+

2z

---------l-i

t

(I + z)"

then f(z) e S. References

tu

and and P. T. Mocanu, Differential Subordinations: Theory York, New Dekker, Marcel 225, Applications, Pure and Applied Mathematics, No. S. S. Miller 2000.

I2l

conditions for M. Nunokawa, s. owa, N. Takahashi and H. Saitoh, sufficient (2002),1385-1390' caratheodory functions,Inclian J. Pure Appl. Math. 33(9)

t3l

Silverman ' Zeszyty M. Obradovic and N. Tuneski, On the starlike criteria defined (2000)' 59-64' 181(24) Mat. Nauk. Politech.Rzeszowskiej

Department of Computer Applications Sri Venkateswara Collegeof Engineering Sriperumbudur 602 105, India ac.in e-mail: vravi@svce.