Coefficient bounds for p-valent functions

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Applied Mathematics and Computation xxx (2006) xxx–xxx www.elsevier.com/locate/amc

Coeﬃcient bounds for p-valent functions Rosihan M. Ali *, V. Ravichandran, N. Seenivasagan School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday

Abstract Sharp bounds for japþ2 la2pþ1 j and jap+3j are derived for certain p-valent analytic functions. These are applied to obtain Fekete-Szego¨ like inequalities for several classes of functions deﬁned by convolution. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Analytic functions; Starlike functions; Convex functions; p-Valent functions; Subordination; Convolution; Fekete-Szego¨ inequalities

1. Introduction Let Ap be the class of all functions of the form 1 X an z n f ðzÞ ¼ zp þ

ð1Þ

n¼pþ1

which are analytic P1in the open unit disk D :¼ {z : jzj < 1} and let A :¼ A1 . For f(z) given by (1) and g(z) given by gðzÞ ¼ zp þ n¼pþ1 bn zn , their convolution (or Hadamard product), denoted by f * g, is deﬁned by 1 X an bn z n : ðf gÞðzÞ :¼ zp þ n¼pþ1

The function f(z) is subordinate to the function g(z), written f(z) g(z), provided there is an analytic function w(z) deﬁned on D with w(0) = 0 and jw(z)j < 1 such that f(z) = g(w(z)). Let u be an analytic function with positive real part in the unit disk D with u(0) = 1 and u 0 (0) > 0 that maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. We deﬁne the class S b;p ðuÞ to be the subclass of Ap consisting of functions f(z) satisfying

*

Corresponding author. E-mail addresses: [email protected] (R.M. Ali), [email protected] (V. Ravichandran), [email protected] (N. Seenivasagan).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.100

Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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1 1 zf 0 ðzÞ 1 uðzÞ; 1þ b p f ðzÞ As special cases, let S p ðuÞ :¼ S 1;p ðuÞ;

ðz 2 D and b 2 C n f0gÞ:

S b ðuÞ :¼ S b;1 ðuÞ;

S ðuÞ :¼ S 1;1 ðuÞ:

For a ﬁxed analytic function g 2 Ap with positive coeﬃcients, deﬁne the class S b;p;g ðuÞ to be the class of all functions f 2 Ap satisfying f g 2 S b;p ðuÞ. ThisPclass includes as special cases several other classes studied 1 in the literature. For example, when gðzÞ ¼ zp þ n¼pþ1 np zn , the class S b;p;g ðuÞ reduces to the class Cb,p(u) consisting of functions f 2 Ap satisfying 1 1 zf 00 ðzÞ 1þ 0 1 þ uðzÞ; ðz 2 D and b 2 C n f0gÞ: b bp f ðzÞ The classes S*(u) and C(u) :¼ C1,1(u) were introduced and studied by Ma and Minda [9]. Deﬁne the class Rb,p(u) to be the class of all functions f 2 Ap satisfying 1 f 0 ðzÞ 1þ 1 uðzÞ; ðz 2 D and b 2 C n f0gÞ; b pzp1 and for a ﬁxed function g with positive coeﬃcients, let Rb,p,g(u) be the class of all functions f 2 Ap satisfying f * g 2 Rb,p(u). Several authors [4,8,10,16,17,12] have studied the classes of analytic functions deﬁned by using the expres0 2 00 ðzÞ sion zff ðzÞ þ a z ffðzÞðzÞ. We shall also consider a class deﬁned by the corresponding quantity for p-valent functions. Deﬁne the class S p ða; uÞ to be the class of all functions f 2 Ap satisfying 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ þ uðzÞ p f ðzÞ p f ðzÞ

ðz 2 D and a P 0Þ:

Note that S p ð0; uÞ is the class S p ðuÞ. Let S p;g ða; uÞ be the class of all functions f 2 Ap for which f g 2 S p ða; uÞ. We shall also consider the class LM p ða; uÞ consisting of p-valent a-convex functions with respect to u. These are functions f 2 Ap satisfying 1 a zf 0 ðzÞ a zf 00 ðzÞ þ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ p f ðzÞ Further let Mp(a, u) be the class of functions f 2 Ap satisfying a 1a 1 zf 0 ðzÞ zf 00 ðzÞ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ f ðzÞ Functions in this class are called logarithmic p-valent a-convex functions with respect to u. In this paper, we obtain Fekete-Szego¨ inequalities and bounds for the coeﬃcient ap+3 for the classes S p ðuÞ and S p;g ðuÞ. These results are then extended to the other classes deﬁned earlier. See [1–7,9,13,14,18] for FeketeSzego¨ problem for certain related classes of functions. Let X be the class of analytic functions of the form wðzÞ ¼ w1 z þ w2 z2 þ

ð2Þ

in the unit disk D satisfying the condition jw(z)j < 1. We need the following lemmas to prove our main results. Lemma 1. If w 2 X, then 8 > < t if t 6 1; 2 jw2 tw1 j 6 1 if 1 6 t 6 1; > : t if t P 1: When t < 1 or t > 1, equality holds if and only if w(z) = z or one of its rotations. If 1 < t < 1, then equality holds if and only if w(z) = z2 or one of its rotations. Equality holds for t = 1 if and only if wðzÞ ¼ Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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kþz kþz z 1þkz ð0 6 k 6 1Þ or one of its rotations while for t = 1, equality holds if and only if wðzÞ ¼ z 1þkz ð0 6 k 6 1Þ or one of its rotations. Also the sharp upper bound above can be improved as follows when 1 < t < 1:

jw2 tw21 j þ ðt þ 1Þjw1 j2 6 1 ð1 < t 6 0Þ and jw2 tw21 j þ ð1 tÞjw1 j2 6 1 ð0 < t < 1Þ: Lemma 1 is a reformulation of a Lemma of Ma and Minda [9]. Lemma 2 [5, inequality 7, p. 10]. If w 2 X, then, for any complex number t, jw2 tw21 j 6 maxf1; jtjg: The result is sharp for the functions w(z) = z2 or w(z) = z. Lemma 3 [11]. If w 2 X, then for any real numbers q1 and q2, the following sharp estimate holds: jw3 þ q1 w1 w2 þ q2 w31 j 6 H ðq1 ; q2 Þ;

ð3Þ

where 8 1 > > > > > > > > > jq2 j > > > > > > > 12 < jq1 jþ1 2 H ðq1 ; q2 Þ ¼ 3 ðjq1 j þ 1Þ 3ðjq1 jþ1þq2 Þ > > > > 2 2 12 > > q1 4 q1 4 1 > > q > 3 2 q21 4q2 3ðq2 1Þ > > > > > 12 > > jq1 j1 : 2 ðjq j 1Þ 3

1

3ðjq1 j1q2 Þ

for ðq1 ; q2 Þ 2 D1 [ D2 ; for ðq1 ; q2 Þ 2

7 S

Dk ;

k¼3

for ðq1 ; q2 Þ 2 D8 [ D9 ; for ðq1 ; q2 Þ 2 D10 [ D11 f2; 1g; for ðq1 ; q2 Þ 2 D12 :

The extremal functions, up to rotations, are of the form wðzÞ ¼ z3 ;

wðzÞ ¼ z;

wðzÞ ¼ w1 ðzÞ ¼ je1 j ¼ je2 j ¼ 1;

zðt1 zÞ ; 1 t1 z

wðzÞ ¼ w0 ðzÞ ¼

zð½ð1 kÞe2 þ ke1 e1 e2 zÞ ; 1 ½ð1 kÞe1 þ ke2 z

wðzÞ ¼ w2 ðzÞ ¼ ih0

zðt2 þ zÞ ; 1 þ t2 z ih0

e1 ¼ t0 e 2 ða bÞ; e2 ¼ e 2 ðia bÞ; rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h0 h0 ba ; b ¼ 1 t20 sin2 ; k¼ a ¼ t0 cos ; 2b 2 2 1 12 2q2 ðq21 þ 2Þ 3q21 2 jq1 j þ 1 t0 ¼ ; t1 ¼ ; 3ðjq1 j þ 1 þ q2 Þ 3ðq2 1Þðq21 4q2 Þ 12 jq1 j 1 h0 q1 q2 ðq21 þ 8Þ 2ðq21 þ 2Þ t2 ¼ ; cos ¼ : 3ðjq1 j 1 q2 Þ 2 2 2q2 ðq21 þ 2Þ 3q21 Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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The sets Dk, k = 1, 2, . . . , 12, are defined as follows:

1 D1 ¼ ðq1 ; q2 Þ : jq1 j 6 ; jq2 j 6 1 ; 2

1 4 D2 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ 6 q2 6 1 ; 2 27

1 D3 ¼ ðq1 ; q2 Þ : jq1 j 6 ; q2 6 1 ; 2

1 2 D4 ¼ ðq1 ; q2 Þ : jq1 j P ; q2 6 ðjq1 j þ 1Þ ; 2 3 D5 ¼ fðq1 ; q2 Þ : jq1 j 6 2; q2 P 1g;

1 D6 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; q2 P ðq21 þ 8Þ ; 12

2 D7 ¼ ðq1 ; q2 Þ : jq1 j P 4; q2 P ðjq1 j 1Þ ; 3

1 2 4 D8 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ 6 q2 6 ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ ; 2 3 27

2 2jq jðjq j þ 1Þ D9 ¼ ðq1 ; q2 Þ : jq1 j P 2; ðjq1 j þ 1Þ 6 q2 6 2 1 1 ; 3 q1 þ 2jq1 j þ 4

2jq1 jðjq1 j þ 1Þ 1 2 6 q2 6 ðq1 þ 8Þ ; D10 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; 2 q1 þ 2jq1 j þ 4 12

2jq jðjq j þ 1Þ 2jq jðjq j 1Þ 6 q2 6 2 1 1 D11 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 ; q1 þ 2jq1 j þ 4 q1 2jq1 j þ 4

2jq jðjq j 1Þ 2 6 q2 6 ðjq1 j 1Þ : D12 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 q1 2jq1 j þ 4 3 2. Coeﬃcient bounds By making use of the Lemmas 1–3, we prove the following bounds for the class S p;g ðuÞ: Theorem 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and r1 :¼

B2 B1 þ pB21 ; 2pB21

r2 :¼

B2 þ B1 þ pB21 ; 2pB21

If f(z) given by (1) belongs to S p ðuÞ, then

8p 2 > < 2 B2 þ ð1 2lÞpB1 apþ2 la2pþ1 6 pB2 1 >

: p 2 B2 þ ð1 2lÞpB21

r3 :¼

B2 þ pB21 : 2pB21

if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :

Further, if r1 6 l 6 r3, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2 If r3 6 l 6 r2, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2

ð4Þ

ð5Þ

ð6Þ

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For any complex number l,

pB B2 apþ2 la2pþ1 6 1 max 1; þ ð1 2lÞpB1 : 2 B1 Further, pB japþ3 j 6 1 H ðq1 ; q2 Þ; 3 where H(q1, q2) is as defined in Lemma 3, q1 :¼

4B2 þ 3pB21 2B1

and

q2 :¼

5

ð7Þ

ð8Þ

2B3 þ 3pB1 B2 þ p2 B31 : 2B1

These results are sharp. Proof. If f ðzÞ 2 S p ðuÞ; then there is an analytic function w(z) = w1z + w2z2 + 2 X such that zf 0 ðzÞ ¼ uðwðzÞÞ: pf ðzÞ Since zf 0 ðzÞ apþ1 ¼1þ zþ pf ðzÞ p

ð9Þ ! a2pþ1 2apþ2 2 þ z þ p p

! a3pþ1 3 3apþ3 3 apþ1 apþ2 þ z þ ; p p p

we have from (9), apþ1 ¼ pB1 w1 ; 1 pB1 w2 þ pðB2 þ pB21 Þw21 apþ2 ¼ 2 and apþ3

ð10Þ ð11Þ

pB1 4B2 þ 3pB21 2B3 þ 3pB1 B2 þ p2 B31 3 w3 þ ¼ w1 w2 þ w1 : 3 2B1 2B1

ð12Þ

Using (10) and (11), we have pB ð13Þ apþ2 la2pþ1 ¼ 1 w2 vw21 ; 2 where B2 v :¼ pB1 ð2l 1Þ : B1 The results (4)–(6) are established by an application of Lemma 1, inequality (7) by Lemma 2 and (8) follows from Lemma 3. To show that the bounds in (4)–(6) are sharp, we deﬁne the functions Kun (n = 2, 3, . . .) by zK 0un ðzÞ ¼ uðzn1 Þ; pK un ðzÞ

0

K un ð0Þ ¼ 0 ¼ ½K un ð0Þ 1

and the functions Fk and Gk (0 6 k 6 1) by zF 0k ðzÞ zðz þ kÞ ¼u ; F k ð0Þ ¼ 0 ¼ F 0k ð0Þ 1 pF k ðzÞ 1 þ kz and zG0k ðzÞ zðz þ kÞ ¼u ; Gk ð0Þ ¼ 0 ¼ G0k ð0Þ 1: pGk ðzÞ 1 þ kz Clearly the functions K un ; F k ; Gk 2 S p ðuÞ. We shall also write Ku :¼ Ku2. Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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If l < r1 or l > r2, then equality holds if and only if f is Ku or one of its rotations. When r1 < l < r2, then equality holds if and only if f is Ku3 or one of its rotations. If l = r1 then equality holds if and only if f is Fk or one of its rotations. Equality holds for l = r2 if and only if f is Gk or one of its rotations. h Corollary 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼

g2pþ1 B2 B1 þ pB21 ; gpþ2 2pB21

r2 :¼

g2pþ1 B2 þ B1 þ pB21 ; gpþ2 2pB21

If f(z) given by (1) belongs to S p;g ðuÞ, then 8 gpþ2 > p 2 > > > 2gpþ2 B2 þ 1 2 g2pþ1 l B1 > > < pB1 apþ2 la2pþ1 6 2gpþ2 > > > > gpþ2 > p > B þ 1 2 l B21 : 2gpþ2 2 g2

r3 :¼

g2pþ1 B2 þ pB21 : gpþ2 2pB21

if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :

pþ1

Further, if r1 6 l 6 r3, then apþ2 la2pþ1 þ

g2pþ1 2gpþ2 pB1

B2 1 þ B1

! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2

If r3 6 l 6 r2, then apþ2 la2pþ1 þ

g2pþ1 2gpþ2 pB1

B2 1þ B1

! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2

For any complex number l, ( B pB 2 1 max 1; þ apþ2 la2pþ1 6 B1 2gpþ2

! ) gpþ2 1 2 2 l B1 : gpþ1

Further, japþ3 j 6

pB1 H ðq1 ; q2 Þ; 3gpþ3

ð14Þ

where H(q1, q2) is as defined in Lemma 3, q1 :¼

4B2 þ 3pB21 2B1

and

q2 :¼

2B3 þ 3pB1 B2 þ p2 B31 : 2B1

These results are sharp. Theorem 2. Let u be as in Theorem 1. If f(z) given by (1) belongs to S b;p;g ðuÞ, then, for any complex number l, we have ( ! ) B pjbjB g2pþ1 2 1 2 max 1; þ bp 1 2 l B1 : apþ2 lapþ1 6 B1 2gpþ2 gpþ2 The result is sharp. Proof. The proof is similar to the proof of Theorem 1. h Proceeding similarly, we now obtain coeﬃcient bounds for the class Rb,p,g(u). Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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Theorem 3. Let u(z) = 1 + B1z + B2z2 + B3z3 + If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, ð15Þ apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼ l

pbB1 ðp þ 2Þ ðp þ 1Þ

2

B2 B1

and

c :¼

bpB1 : 2þp

Further, japþ3 j 6

jbjpB1 H ðq1 ; q2 Þ; 3þp

ð16Þ

where H(q1, q2) is as defined in Lemma 3, q1 :¼

2B2 B1

and

q2 :¼

B3 : B1

These results are sharp. Proof. A computation shows that 1 f 0 ðzÞ pþ1 pþ2 pþ3 1þ apþ1 z þ apþ2 z2 þ apþ3 z3 þ : 1 ¼1þ p1 b pz bp bp bp Thus bpB1 apþ2 la2pþ1 ¼ w2 vw21 ¼ c w2 vw21 ; pþ2 h i 1 lð2þpÞ 1 where v :¼ bpBðpþ1Þ BB21 and c :¼ bpB . The result now follows from Lemmas 2 and 3. 2 pþ2

ð17Þ h

Remark 1. When p = 1 and uðzÞ ¼

1 þ Az 1 þ Bz

ð1 6 B 6 A 6 1Þ;

inequality (15) reduces to give the inequality [3, Theorem 4, p. 894]. For the class Rb,p,g(u), we have the following result. Theorem 4. Let u(z) = 1 + B1z + B2z2 + B3z3 + . If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼

lg2pþ1 pbB1 ðp þ 2Þ B2 gpþ2 ðp þ 1Þ2 B1

and

c :¼

bpB1 : gpþ2 ð2 þ pÞ

Further, japþ3 j 6

jbjpB1 H ðq1 ; q2 Þ; ð3 þ pÞgpþ3

ð18Þ

where H(q1, q2) is as defined in Lemma 3, q1 :¼

2B2 B1

and

q2 :¼

B3 : B1

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These results are sharp. We now prove the coeﬃcient bounds for S p;g ða; uÞ. Theorem 5. Let u(z) = 1 + B1z + B2z2 + B3z3 + Further let r1 :¼

ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 ; 2pB21

r2 :¼

ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 ; 2pB21

ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 ; 2pB21 2lpB1 ðpa þ 2a þ 1Þ pB1 B2 ; v :¼ 2 1 þ aðp þ 1Þ B1 ð1 þ aðp þ 1ÞÞ

r3 :¼

If f(z) given by (1) belongs to 8 > < cv 2 a c la 6 pþ2 pþ1 > : cv

c :¼

pB1 : 2ð1 þ aðp þ 2ÞÞ

S p ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ;

ð19Þ

if l P r2 :

Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 c 2 ðv þ 1Þjapþ1 j 6 c: apþ2 la2pþ1 þ p2 B21 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 c 2 ð1 vÞjapþ1 j 6 c: apþ2 la2pþ1 þ 2 2 p B1 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg:

ð20Þ

Further, pB1 H ðq1 ; q2 Þ; að5 p p2 Þ þ 3 where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ q1 :¼ þ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ þ : q2 :¼ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 These results are sharp. japþ3 j 6

ð21Þ

Proof. If f ðzÞ 2 S p ða; uÞ. It is easily shown that 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ ð1 þ aðp þ 1ÞÞ ð1 þ aðp þ 2ÞÞ ð1 þ aðp þ 1ÞÞ 2 þ :¼ 1 þ apþ1 z þ 2apþ2 apþ1 z2 p f ðzÞ p f ðzÞ p p p ð3 þ að5 p p2 ÞÞ ð3 þ að3p þ 5ÞÞ ð1 þ aðp þ 1ÞÞ 3 apþ3 apþ1 apþ2 þ apþ1 z3 þ p p p þ

ð22Þ

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The proof can now be completed as in the proof of Theorem 1. For the class

S p;g ða; uÞ,

9

h

we have the following result.

Theorem 6. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼ r2 :¼ r3 :¼ v :¼

g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 2pB21 gpþ2

2gpþ2 lpB1 ðpa þ 2a þ 1Þ g2pþ1 ð1

þ aðp þ 1ÞÞ

If f(z) given by (1) belongs to 8 > < cv apþ2 la2pþ1 6 c > : cv

2

; ;

;

pB1 B2 ; 1 þ aðp þ 1Þ B1

c :¼

pB1 : 2gpþ2 ð1 þ aðp þ 2ÞÞ

S p;g ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :

Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 2 ðv þ 1Þjapþ1 j 6 c: apþ2 la2pþ1 þ p2 B21 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 ð1 vÞjapþ1 j2 6 c: apþ2 la2pþ1 þ p2 B21 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg: Further, japþ3 j 6

pB1 H ðq1 ; q2 Þ; gpþ3 ½að5 p p2 Þ þ 3

ð23Þ

where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ q1 :¼ þ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and q2 :¼

B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ þ : 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1

These results are sharp. Remark 2. When p = 1 and uðzÞ ¼

1 þ zð1 2bÞ 1z

ða P 0; 0 6 b < 1Þ;

(19) and (20) of Theorem 5 reduces to [4, Theorem 4 and 3, p. 95]. Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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For the class LM p ða; uÞ, we now get the following coeﬃcient bounds: Theorem 7. Let u(z) = 1 + B1z + B2z2 + B3z3 + . Let c3 ðB2 B1 Þ þ c2 B21 ; c1 B21

r1 :¼

c1 :¼ 4p2 ðp þ 2 2aÞ; v :¼

c 1 l c 2 B2 ; c3 B1

T 2 :¼

r2 :¼

c3 ðB2 þ B1 Þ þ c2 B21 ; c1 B21 2

r3 :¼

c2 :¼ ð1 aÞ½2pðp þ 1Þ þ a þ 2ap3 ; T 1 :¼

c3 B2 þ c2 B21 ; c1 B21 2

c3 :¼ 2ðp þ 1 aÞ ;

3ðp2 þ ð1 aÞð3p þ 2ÞÞ ; p3

1a a ½ðp þ 1Þ3 ½6pðp þ aÞ þ aða þ 1Þ pa½6p3 þ p2 ð1 þ 7aÞ þ 3pð1 þ 3aÞ þ 3a þ : 6p6 p

If f(z) given by (1) belongs to LM p ða; uÞ, then 8 p2 B1 v > > > 2ðpþ22aÞ if l 6 r1 ; < p2 B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ22aÞ > > > 2 : p B1 v if l P r2 : 2ðpþ22aÞ Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 þ vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2 2aÞ

ð24Þ

ð25Þ

ð26Þ

ð27Þ

Further, japþ3 j 6

p 2 B1 H ðq1 ; q2 Þ; 3ðp 3a þ 3Þ

where H(q1, q2) is as defined in Lemma 3, q1 :¼

2B2 T 1 p 4 B1 B1 2ðp a þ 1Þðp 2a þ 2Þ

q2 :¼

B3 T 1 p4 ðc3 B2 þ c2 B21 Þ T 2 p6 B21 : B1 2c3 ðp a þ 1Þðp 2a þ 2Þ ðp a þ 1Þ3

and

These results are sharp. Proof. The proof is similar to the proof of Theorem 1. h Remark 3. As special cases, we note that for p = 1, inequalities (24)–(27) in Theorem 7 are those found in [15, Theorems 2.1 and 2.2, p. 3]. Additionally, if a = 1, then inequalities (24)–(26) are the results established in [14, Theorem 2.1, Remark 2.2, p. 3]. Our ﬁnal result is on the coeﬃcient bounds for Mp(a, u). Please cite this article in press as: R.M. Ali et al., Coeﬃcient bounds for p-valent functions, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.08.100

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11

Theorem 8. Let u(z) = 1 + B1z + B2z2 + B3z3 + Let r1 :¼

c1 ðB2 B1 Þ þ c2 B21 ; c3 B21

r2 :¼

c1 ðB2 þ B1 Þ þ c2 B21 ; c3 B21

2

c2 :¼ a þ pðp þ 2aÞ; c1 :¼ pð1 þ paÞ ; ½pðp þ 2aÞð2l 1Þ aB1 B2 : v :¼ c3 B1

r3 :¼

c1 B2 þ c2 B21 ; c3 B21

c3 :¼ 2pðp þ 2aÞ;

If f(z) given by (1) belongs to Mp(a, u), then 8 2 p B1 v > > > 2ðpþ2aÞ if l 6 r1 ; < p2 B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ2aÞ > > > 2 : p B1 v if l P r : 2 2ðpþ2aÞ

ð28Þ

Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 þ vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ

ð29Þ

If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ

ð30Þ

For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2aÞ

ð31Þ

Further, japþ3 j 6

p 2 B1 H ðq1 ; q2 Þ; 3ðp þ 3aÞ

where H(q1, q2) is as defined in Lemma 3, 2B2 3ðp2 þ 3pa þ 2aÞ B1 ; þ 2ð1 þ paÞðp þ 2aÞ B1 B3 3ðp2 þ 3pa þ 2aÞ ðp4 þ 5ap3 þ 3p2 að2a þ 1Þ þ pað9a 2Þ þ 2a2 Þ 2 B2 þ q2 :¼ þ B1 : 3 B1 2ð1 þ paÞðp þ 2aÞ 2pð1 þ paÞ ðp þ 2aÞ

q1 :¼

These results are sharp. Proof. For f(z) 2 Mp(a, u), a computation shows that ! ðp2 þ 2pa þ aÞa2pþ1 2ðp þ 2aÞapþ2 2 1 a zf 0 ðzÞ a zf 00 ðzÞ ð1 þ paÞapþ1 þ ¼1þ zþ z þ 1þ 0 p p3 p2 p f ðzÞ p f ðzÞ 3ðp þ 3aÞ 3ðp2 þ 3pa þ 2aÞ p3 þ 3p2 a þ 3pa þ a 3 þ apþ3 apþ1 apþ2 þ apþ1 z3 p2 p3 p4 þ

The remaining part of the proof is similar to the proof of Theorem 1.

h

b

1þz Remark 4. When p = 1 and uðzÞ ¼ ð1z Þ ða P 0; 0 < b 6 1Þ; (28) and (31) of Theorem 8 reduce to [2, Theorems 2.1 and 2.2, p. 23]. When p = 1 and a = 1, (28)–(30) of Theorem 8 reduce to [9, Theorem 3 and Remark, p. 7].

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Acknowledgement The authors gratefully acknowledged support from the research grant IRPA 09-02-05-00020 EAR. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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FORMULAS FOR EXAM 1 1) Coefficient of variation ...