6. A Class of Multivalent Analytic Functions with fixed Argument

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Volume 2 Issue 1(2010), Pages 56-65.

A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH FIXED ARGUMENT OF COEFFICIENTS INVOLVING WRIGHT’S GENERALIZED HYPERGEOMETRIC FUNCTIONS

POONAM SHARMA

Abstract. In this paper, a class of analytic functions with fixed argument of its coefficients involving Wright’s generalized hypergeometric function is defined with the help of subordination. The coefficient inequalities have been derived. Growth, distortion bounds and extreme points for functions belonging to the defined class have been investigated with consequent results.

1. _{Introduction and Preliminaries} Let A𝑚denote a class of functions𝑓(𝑧) of the form:

𝑓(𝑧) =𝑧𝑚+

∞ ∑

𝑘=1

𝑎𝑚+𝑘𝑧𝑚+𝑘 (𝑎𝑚+𝑘 ∈C, 𝑚∈N ={1,2,3...}), (1.1)

which are analytic in an open unit disk U := {𝑧 : 𝑧 ∈ C and ∣𝑧∣ < 1} and its subclass is denoted by A𝜃_𝑚 whose members are of the form:

𝑓(𝑧) =𝑧𝑚+𝑒𝑖𝜃

∞ ∑

𝑘=1

∣𝑎𝑚+𝑘∣𝑧𝑚+𝑘 (𝑚∈N ={1,2,3...}), (1.2)

where𝜃 is the ﬁxed argument of𝑎𝑚+𝑘 ∕= 0 (𝑘≥1).Note that A1≡A.

A function𝑓(𝑧)∈A𝑚is said to be in the class S∗𝑚(𝛼) if it satisﬁes

Re

{

𝑧𝑓′(𝑧) 𝑓(𝑧)

}

> 𝛼,(0≤𝛼 <1, 𝑧∈U)

and functions therein are called starlike of order𝛼.

The Wright’s (psi) function𝑝Ψ𝑞(𝑧) is a generalized hypergeometric function [12] whose series representation is given by

2000Mathematics Subject Classiﬁcation. Primary 30C45, 30C55; Secondary 26A33, 33C60. Key words and phrases. Analytic functions; starlike functions; Wright’s generalized hypergeo-metric function; subordination.

c

⃝2010 Universiteti i Prishtinës, Prishtinë, Kosovë.

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𝑝Ψ𝑞((𝑎1, 𝐴1), ...,(𝑎𝑝, 𝐴𝑝); (𝑏1, 𝐵1), ...,(𝑏𝑞, 𝐵𝑞);𝑧) =

∞ ∑

𝑘=0

𝑝

∏

𝑖=1

Γ(𝑎𝑖+𝐴𝑖𝑘)𝑧𝑛

𝑞

∏

𝑖=1

Γ(𝑏𝑖+𝐵𝑖𝑘)𝑘!

, (1.3)

where 𝑎𝑖(𝑖 = 1,2, ..., 𝑝), 𝑏𝑖(𝑖 = 1,2, ..., 𝑞) are positive real numbers and 𝐴𝑖(𝑖 =

1,2, ..., 𝑝), 𝐵𝑖(𝑖= 1,2, ..., 𝑞) are positive integers such that 1 + 𝑞

∑

𝑖=1

𝐵𝑖− 𝑝

∑

𝑖=1

𝐴𝑖 ≥0.

Taking 𝐴𝑖 = 1(𝑖 = 1,2, ..., 𝑝), 𝐵𝑖 = 1(𝑖 = 1,2, ..., 𝑞), we see that

𝑞 ∏

𝑖=1 Γ(𝑏𝑖)

𝑝 ∏

𝑖=1 Γ(𝑎𝑖)

𝑝Ψ𝑞(𝑧)

reduces to a familiar generalized hypergeometric function:

𝑝F𝑞(𝑧)≡𝑝F𝑞(𝑎1, ..., 𝑎𝑝;𝑏1, ..., 𝑏𝑞;𝑧) =

∞ ∑

𝑘=0

(𝑎1)𝑘....(𝑎𝑝)𝑘𝑧𝑘

(𝑏1)𝑘...(𝑏𝑞)𝑘𝑘!

, (1.4)

where the Pochhammer symbol is deﬁned by (𝑎)𝑘 =

Γ(𝑎+𝑘)

Γ(𝑎) for non-negative integrs

𝑘.

Similar to the operator deﬁned earlier by Dziok and Raina [4] (see also [1, 2], [6] and [5]), involving psi function 𝑞+1Ψ𝑞(𝑧) for positive real numbers 𝑎𝑖, 𝑏𝑖(𝑖 =

1,2, ..., 𝑞) and for positive integers𝐴𝑖, 𝐵𝑖(𝑖= 1,2, ..., 𝑞) such that 𝑞

∑

𝑖=1

(𝐵𝑖−𝐴𝑖)≥0,

an operator𝐼𝑞

𝑚([𝑎1])𝑓 ≡𝐼𝑚𝑞 ((𝑎𝑖, 𝐴𝑖)1,𝑞,(𝑏𝑖, 𝐵𝑖)1,𝑞)𝑓 : A𝑚→A𝑚 is deﬁned as:

𝐼_𝑚𝑞 ([𝑎1])𝑓 =𝑧𝑚

𝑞

∏

𝑖=1

Γ(𝑏𝑖) Γ(𝑎𝑖)

𝑞+1Ψ𝑞(𝑧)∗𝑓, (1.5)

where

𝑞+1Ψ𝑞(𝑧)≡𝑞+1Ψ𝑞((𝑎1, 𝐴1), ...,(𝑎𝑞, 𝐴𝑞),(1,1); (𝑏1, 𝐵1), ...,(𝑏𝑞, 𝐵𝑞);𝑧).

Let 𝑓(𝑧)∈ A𝑚 be of the form (1.1), then the series expansion of 𝐼𝑚𝑞 ([𝑎1])𝑓 is

given by

𝐼_𝑚𝑞 ([𝑎1])𝑓 =𝑧𝑚+ ∞ ∑

𝑘=1

𝜃𝑚+𝑘𝑎𝑚+𝑘𝑧𝑚+𝑘, (1.6)

where

𝜃𝑚+𝑘 = 𝑞

∏

𝑖=1

Γ(𝑎𝑖+𝐴𝑖𝑘)Γ(𝑏𝑖) Γ(𝑏𝑖+𝐵𝑖𝑘)Γ(𝑎𝑖)

, 𝑘≥1. (1.7)

The series expansion of𝐼𝑞

𝑚([𝑎1+ 1])𝑓 ≡𝐼𝑚[(𝑎1+ 1, 𝐴1),(𝑎𝑖, 𝐴𝑖)2,𝑞,(𝑏𝑖, 𝐵𝑖)1,𝑞]𝑓 : A𝑚→A𝑚is given by

𝐼_𝑚𝑞 ([𝑎1+ 1])𝑓 =𝑧𝑚+ ∞ ∑

𝑘=1

𝜃+_𝑚₊_𝑘𝑎𝑚+𝑘𝑧𝑚+𝑘 (1.8)

with

𝜃+_𝑚₊_𝑘 = (𝑎1+𝐴1𝑘) 𝑎1

𝜃𝑚+𝑘.

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𝐴1𝑧(𝐼𝑚𝑞 ([𝑎1])𝑓)′=𝑎1𝐼𝑚𝑞 ([𝑎1+ 1])𝑓 −(𝑎1−𝑚𝐴1)𝐼𝑚𝑞 ([𝑎1])𝑓 (1.9)

which shows that for 𝑎𝑖 =𝑚𝐴𝑖, 𝐼𝑚𝑞 ([𝑎1+ 1])𝑓 =

𝑧(𝐼𝑞 𝑚([𝑎1])𝑓)′

𝑚 . Note that𝐼 𝑞 𝑚𝑓 ≡𝑓 if 𝐴𝑖 =𝐵𝑖, 𝑎𝑖=𝑏𝑖 (𝑖= 1,2, ..., 𝑞).

With the use of subordination, classes in which the linear operator 𝐼₁𝑞([𝑎1])𝑓

is involved, are deﬁned and studied in [2] and [6]. Involving 𝐼𝑞

𝑚([𝑎1])𝑓 in [9], a

class of functions 𝑓 ∈ A𝜃

𝑚 with the help of subordination is deﬁned and studied. Motivated with the work of Raina [9], involving 𝐼𝑞

𝑚([𝑎1])𝑓 and 𝐼𝑚𝑞 ([𝑎1+ 1])𝑓 ,

we deﬁne a class for functions𝑓 ∈A𝑚 as follows:

Deﬁnition 1.1. Let A𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)denote a class of functions𝑓 ∈A𝑚

satis-fying

(1−𝜆)𝐼 𝑞

𝑚([𝑎1])𝑓

𝑧𝑚 +𝜆

𝐼𝑞

𝑚([𝑎1+ 1])𝑓

𝑧𝑚 ≺

1 +𝐴𝑧

1 +𝐵𝑧, (1.10)

where 0 ≤ 𝜆 ≤1, and −1 ≤ 𝐴 < 𝐵 ≤ 1,0 ≤ 𝐵. Denote A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆) ≡

A𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)∩A𝜃𝑚.

For positive real𝑎and for positive integer𝐴,we have [[11], 240, Eq. (1.26)]:

Γ(𝑎+𝑘𝐴) = Γ(𝑎)(𝑎 𝐴

)

𝑘

(_𝑎_{+ 1}

𝐴

)

𝑘 ...

(_𝑎₊_𝐴 −1 𝐴

)

𝑘

(𝐴)𝑘𝐴, 𝑘= 0,1,2, ...

when it is used together with the result [[3], p.57]:

Γ(𝑎+𝑘)Γ(𝑏+𝑘) Γ(𝑐+𝑘)Γ(𝑑+𝑘) =𝑘

𝑎+𝑏−𝑐−𝑑

[

1 +𝑂

(₁

𝑘

)]

, 𝑘= 1,2,3, ...

we obtain that for positive real numbers𝑎𝑖, 𝑏𝑖(𝑖= 1,2, ..., 𝑞) and positive integers

𝐴𝑖, 𝐵𝑖(𝑖 = 1,2, ..., 𝑞) such that 𝑞

∑

𝑖=1

(𝐵𝑖−𝐴𝑖)≥0, the series

∞ ∑

𝑘=1

𝜃𝑚+𝑘,where 𝜃𝑚+𝑘

is given by (1.7), converges absolutly if

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1 + 1 2

𝑞

∑

𝑖=1

(𝐴𝑖−𝐵𝑖). (1.11)

For details one may refer to [8] and [10].

The purpose of this paper is to investigate the coeﬃcient inequalities, growth and distortion bounds with certain conditions on parameters and extreme points for functions belonging to the class A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆). Some consequent results

are also mentioned.

2. _{Coefficient Inequalities}

Theorem 2.1. Let 𝑓(𝑧) ∈ A𝜃_𝑚 of the form (1.2) belong to A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆),

then

∞ ∑

𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣<

𝑎1(𝐵−𝐴) [√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}], (2.1)

where 𝜃𝑚+𝑘 is given by (1.7)and 0 ≤𝜆≤1, −1 ≤𝐴 < 𝐵 ≤1,0 ≤𝐵 ,𝜃 is the

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Proof. Let a function𝑓belong to the class A𝜃𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆), then from Deﬁnition

1.1, we have

(1−𝜆)𝐼 𝑞

𝑚([𝑎1])𝑓

𝑧𝑚 +𝜆

𝐼𝑞

𝑚([𝑎1+ 1])𝑓

𝑧𝑚 =

1 +𝐴𝑤(𝑧) 1 +𝐵𝑤(𝑧), where𝑤(0) = 0 and∣𝑤(𝑧)∣<1 for 𝑧∈U.Hence we have

∣𝑤(𝑧)∣=

𝑧𝑚₋₍₁₋_𝜆₎_𝐼𝑞

𝑚([𝑎1])𝑓−𝜆𝐼𝑚𝑞 ([𝑎1+ 1])𝑓

𝐵[(1−𝜆)𝐼𝑚𝑞 ([𝑎1])𝑓+𝜆𝐼𝑚𝑞 ([𝑎1+ 1])𝑓]−𝐴𝑧𝑚

<1, 𝑧∈U. (2.2)

On using (1.6) and (1.8), inequality (2.2) yields

∞ ∑ 𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣𝑧𝑘

<

𝑎1(𝐵−𝐴) +𝐵𝑒𝑖𝜃 ∞ ∑

𝑘=1

. (2.3)

For 0< 𝑧=𝑟 <1,

∞ ∑

𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣𝑟𝑘 =:𝜂 is real, we get

∣𝜂∣<𝑎1(𝐵−𝐴) +𝐵𝑒𝑖𝜃𝜂

. (2.4)

On solving (2.4) for𝜂,we get

𝜂 <_[√ 𝑎1(𝐵−𝐴)

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}],

which proves the inequality (2.1). _□

The inequality (2.1) is sharp and the extremal function is given by

𝑓𝑘(𝑧) =𝑧𝑚+𝑒𝑖𝜃

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}]𝑧

𝑚+𝑘_{, 𝑘}_≥₁_. _(2.5)

Corollary 2.2. Let𝑓(𝑧)∈A𝜃_𝑚of the form(1.2)belong to A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)then

for

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1 + 1₂ 𝑞 ∑ 𝑖=1 (𝐴𝑖−𝐵𝑖), ∞ ∑ 𝑘=1

∣𝑎𝑚+𝑘∣<

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1)𝜙[ √

where𝜙=min

𝑘≥1{𝜃𝑚+𝑘}, 𝜃𝑚+𝑘 is given by (1.7).

Proof. Under the given hypothesis, convergence of the series

∞ ∑

𝑘=1

𝜃𝑚+𝑘 implies that

𝜃𝑚+𝑘,given in (1.7) is bounded for𝑘≥1. Let𝜙:= min

𝑘≥1{𝜃𝑚+𝑘}, then by Theorem

2.1 we get the result. _□

Corollary 2.3. Let𝑓(𝑧)∈A𝜃_𝑚of the form(1.2)belong to A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)then

for

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1₂ 𝑞 ∑ 𝑖=1 (𝐴𝑖−𝐵𝑖), ∞ ∑ 𝑘=1

(𝑚+𝑘)∣𝑎𝑚+𝑘∣<

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1)𝜑[ √

where𝜑=min

𝑘≥1 {_𝜃

𝑚+𝑘

}

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Proof. Under the given hypothesis, convergence of the series ∑

𝑘=1

𝜃𝑚+𝑘

𝑚+𝑘 implies that

𝜃𝑚+𝑘

𝑚+𝑘 is bounded for 𝑘 ≥ 1, where 𝜃𝑚+𝑘 is given by (1.7) and hence let 𝜑 := min

𝑘≥1 {_𝜃

𝑚+𝑘

}

, then by Theorem 2.1 we get the result. _□

Taking𝜃= (2𝑛−1)𝜋, 𝑛= 1,2,3, ..in Corollary 2.2 we get following result.

Corollary 2.4. Let𝑓(𝑧)∈A(2_𝑚𝑛−1)𝜋of the form(1.2)belong to A(2_𝑚𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆),

then for

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1 +1₂ 𝑞

∑

𝑖=1

(𝐴𝑖−𝐵𝑖),

∞ ∑

𝑘=1

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1)𝜙(1 +𝐵)

,

where𝜙=min

Remark. Raina [9]studied the class of functions 𝑓(𝑧)∈A𝜃_𝑚 and obtained results

assuming𝜃𝑚+𝑘 and 𝜃_𝑚𝑚₊+_𝑘𝑘 to be increasing functions.

Theorem 2.5. A function𝑓(𝑧)∈A(2_𝑚𝑛−1)𝜋of the form(1.2)belong to A(2_𝑚𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)

if and only if

∞ ∑

𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣<

𝑎1(𝐵−𝐴)

(1 +𝐵) , (2.6)

where𝜃𝑚+𝑘 is given by (1.7) and0≤𝜆≤1, −1≤𝐴 < 𝐵≤1,0≤𝐵.

Proof. We need to show only suﬃcient condition for𝑓 ∈A_𝑚(2𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆).

Consider

∞ ∑

𝑘=1

−

𝑎1(𝐵−𝐴)−𝐵 ∞ ∑

𝑘=1

≤ ∞ ∑

𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣ −

[

𝑎1(𝐵−𝐴)−𝐵 ∞ ∑

𝑘=1

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣

]

≤ ∞ ∑

𝑘=1

(1+𝐵)(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣−𝑎1(𝐵−𝐴)<0,

if (2.6) holds, which proves (2.3) for𝜃= (2𝑛−1)𝜋and hence the result.

□

The result is sharp for the function:

𝑓𝑘(𝑧) =𝑧𝑚−

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘(1 +𝐵)

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Corollary 2.6. Let a function 𝑓(𝑧) ∈ A(2𝑚𝑛−1)𝜋 of the form (1.2) belong to

A(2_𝑚𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)then

𝑎1(𝐵−𝐴)

, 𝑘≥1. (2.8)

3. _{Growth and Distortion Bounds}

In this section we ﬁnd growth and distortion bounds for functions belonging to the class A𝜃𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆) with the use of Corollaries 2.2 and 2.3.

Theorem 3.1. Let𝑓(𝑧)∈A𝜃_𝑚of the form(1.2)belong to A𝜃_𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆),then

for0<∣𝑧∣=𝑟 <1 and for

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1 + 1₂ 𝑞

∑

𝑖=1

(𝐴𝑖−𝐵𝑖)

𝑟𝑚−𝑟𝑚+1 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜙

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}] ≤ ∣𝑓(𝑧)∣

≤𝑟𝑚+𝑟𝑚+1 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜙[

√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}], (3.1)

where𝜙=min

Proof. From (1.2), we have

∣𝑓(𝑧)∣ ≤𝑟𝑚+

∞ ∑

𝑘=1

∣𝑎𝑚+𝑘∣𝑟𝑚+𝑘,0<∣𝑧∣=𝑟 <1.

Using Corollary 2.2, we get

∣𝑓(𝑧)∣ ≤𝑟𝑚+𝑟𝑚+1 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜙

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}],0<∣𝑧∣=𝑟 <1.

Similary we get

∣𝑓(𝑧)∣ ≥𝑟𝑚−𝑟𝑚+1 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜙

[√

which completes the proof.

□

Theorem 3.2. Let 𝑓(𝑧) ∈ A𝜃𝑚 of the form (1.2) belong to A 𝜃

𝑚(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆),

then for0<∣𝑧∣=𝑟 <1 and for

𝑞

∑

𝑖=1

(𝑏𝑖−𝑎𝑖)>1₂ 𝑞

∑

𝑖=1

(𝐴𝑖−𝐵𝑖),

𝑚𝑟𝑚−1−𝑟𝑚 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜑

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}] ≤ ∣𝑓 ′₍_𝑧₎_∣

≤𝑚𝑟𝑚−1+𝑟𝑚 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜑

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}], (3.2)

where𝜑=min

𝑘≥1 {_𝜃

𝑚+𝑘

}

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Proof. From (1.2), we have

∣𝑓′(𝑧)∣ ≤𝑚𝑟𝑚−1+

∞ ∑

𝑘=1

(𝑚+𝑘)∣𝑎𝑚+𝑘∣𝑟𝑚+𝑘−1,0<∣𝑧∣=𝑟 <1.

Using Corollary 2.3, we get

∣𝑓′(𝑧)∣ ≤𝑚𝑟𝑚−1+𝑟𝑚 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜑

[√

1−𝐵2_𝑠𝑖𝑛2_𝜃₋_{𝐵𝑐𝑜𝑠𝜃}].

Similary we get

∣𝑓′(𝑧)∣ ≥𝑚𝑟𝑚−1−𝑟𝑚 𝑎1(𝐵−𝐴) (𝑎1+𝜆𝐴1)𝜑

[√

which completes the proof. _□

4. _{Extreme Points}

In this section we ﬁnd extreme points for the class A(2_𝑚𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆).

Theorem 4.1. Let 𝑓(𝑧)∈A(2_𝑚𝑛−1)𝜋 and

𝑓0(𝑧) =𝑧𝑚, 𝑓𝑘(𝑧) =𝑧𝑚−

𝑎1(𝐵−𝐴)

𝑧𝑚+𝑘, 𝑘≥1. (4.1)

Then𝑓(𝑧)∈A𝑚(2𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆) if only if it can be expressed in the form

𝑓(𝑧) =

∞ ∑

𝑘=0

𝜆𝑘𝑓𝑘(𝑧), (4.2)

where 𝜆𝑘 ≥ 0and

∞ ∑

𝑘=0

𝜆𝑘 = 1 and 𝑓𝑘′𝑠 for 𝑘 ≥ 0 are the extreme points for

A(2_𝑚𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆)class.

Proof. Let

𝑓(𝑧) =

∞ ∑

𝑘=0

𝜆𝑘𝑓𝑘(𝑧)

= (1− ∞ ∑

𝑘=1

𝜆𝑘)𝑧𝑚+

∞ ∑

𝑘=1

𝜆𝑘

(

𝑧𝑚− 𝑎1(𝐵−𝐴)

𝑧𝑚+𝑘

)

= 𝑧𝑚− ∞ ∑

𝑘=1

𝜆𝑘

𝑎1(𝐵−𝐴)

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘(1 +𝐵) 𝑧𝑚+𝑘,

which proves that𝑓(𝑧)∈A_𝑚(2𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆).Since by Theorem 2.5 ∞

∑

𝑘=1

(1 +𝐵)(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘 𝑎1(𝐵−𝐴)

[ _𝑎

1(𝐵−𝐴)𝜆𝑘 (𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘(1 +𝐵)

]

=

∞ ∑

𝑘=1

𝜆𝑘<1−𝜆0≤1.

Conversely, suppose 𝑓(𝑧) ∈ A_𝑚(2𝑛−1)𝜋(𝑝,[𝑎1], 𝐴, 𝐵, 𝜆) then using Corollary 2.6,

we set𝜆𝑘 =

(𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘(1+𝐵)

𝑎1(𝐵−𝐴) 𝑎𝑚+𝑘 for𝑘≥1 and𝜆0= 1− ∞ ∑

𝑘=1

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Thus

𝑓(𝑧) = 𝑧𝑚− ∞ ∑

𝑘=1

∣𝑎𝑚+𝑘∣𝑧𝑚+𝑘

= 𝑓0(𝑧)− ∞ ∑

𝑘=1

𝑎1(𝐵−𝐴)𝜆𝑘 (𝑎1+𝜆𝐴1𝑘)𝜃𝑚+𝑘(1 +𝐵)

𝑧𝑚+𝑘

= 𝑓0(𝑧)− ∞ ∑

𝑘=1

[𝑧𝑚−𝑓𝑘(𝑧)]𝜆𝑘

=

∞ ∑

𝑘=0

𝜆𝑘𝑓𝑘(𝑧).

This completes the proof. _□

5. Some Consequences of Main Results

Taking𝑎1=𝑚𝐴1in Theorm 2.1, we get following result.

Corollary 5.1. A function 𝑓(𝑧)∈A(2_𝑚𝑛−1)𝜋 of the form (1.2)satisﬁes

(1−𝜆)𝐼 𝑞

𝑚([𝑎1])𝑓

𝑧𝑚 +𝜆

𝑧(𝐼𝑞

𝑚([𝑎1])𝑓) ′

𝑧𝑚 ≺

1 +𝐴𝑧 1 +𝐵𝑧

if and only if

∞ ∑

𝑘=1

(𝑚+𝜆𝑘)𝜃𝑚+𝑘∣𝑎𝑚+𝑘∣<

𝑚(𝐵−𝐴) (1 +𝐵) ,

where𝜃𝑚+𝑘 is deﬁned in (1.7)and0≤𝜆≤1, −1≤𝐴 < 𝐵≤1,0≤𝐵.

In Corolary 5.1, on replacing𝜃𝑚+𝑘 by𝑏𝑚+𝑘,following result can be obtained.

Corollary 5.2. Let 𝑔(𝑧)∈A𝑚 of the form:

𝑔(𝑧) =𝑧𝑚+

∞ ∑

𝑘=1

𝑏𝑚+𝑘𝑧𝑚+𝑘 (𝑏𝑚+𝑘≥0, 𝑚∈N={1,2,3...})

and0 ≤𝜆≤1, −1≤𝐴 < 𝐵≤1,0 ≤𝐵. A function 𝑓(𝑧)∈A(2𝑚𝑛−1)𝜋 of the form (1.2)satisﬁes

(1−𝜆)(𝑓∗𝑔) 𝑧𝑚 +𝜆

𝑧(𝑓∗𝑔)′

𝑧𝑚 ≺

1 +𝐴𝑧 1 +𝐵𝑧, 𝑧∈U

if and only if

∞ ∑

𝑘=1

(𝑚+𝜆𝑘)𝑏𝑚+𝑘∣𝑎𝑚+𝑘∣<

𝑚(𝐵−𝐴) (1 +𝐵) .

Corollary 5.3. Let 0< 𝜆≤1,−1< 𝐴 <0 , if a function𝑓(𝑧)∈A𝑚satisﬁes

(1−𝜆)𝐼 𝑞

𝑚([𝑎1])𝑓

𝑧𝑚 +𝜆

𝐼𝑞

𝑚([𝑎1+ 1])𝑓

𝑧𝑚 ≺1 +𝐴𝑧,

then𝐼_𝑚𝑞 ([𝑎1])𝑓 ∈S∗𝑚(𝛼),𝛼:=

(

𝑚−_𝜆𝐴2𝑎1∣𝐴∣ 1(1−∣𝐴∣)

)

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Proof. Let𝑝(𝑧) :=𝐼𝑚([𝑎1])𝑓

𝑧𝑚 then with the use of identity (1.9) we have

𝑝(𝑧) +𝜆𝐴1 𝑎1

𝑧𝑝′(𝑧)≺1 +𝐴𝑧.

By a well known Lemma of Hallenbeck and Ruscheweyh [7] we get that𝑝(𝑧)≺1+𝐴𝑧 and hence:

𝐼𝑞

𝑚([𝑎1])𝑓

𝑧𝑚 −1

<∣𝐴∣ and

𝐼𝑞

𝑚([𝑎1])𝑓

𝑧𝑚

>1− ∣𝐴∣,

which with the use of hypothesis and the identity(1.9), derives:

𝑧(𝐼_𝑚𝑞 ([𝑎1])𝑓)′

𝑧𝑚 −𝑚

𝐼_𝑚𝑞 ([𝑎1])𝑓

𝑧𝑚

< 2𝑎1∣𝐴∣ 𝜆𝐴1(1− ∣𝐴∣)

𝐼_𝑚𝑞 ([𝑎1])𝑓

𝑧𝑚

.

That evidently yields:

𝑧(𝐼𝑞 𝑚([𝑎1])𝑓)′

𝐼𝑚𝑞([𝑎1])𝑓 −𝑚 <

2𝑎1∣𝐴∣

𝜆𝐴1(1−∣𝐴∣).

Therefore Re{𝑧(𝐼𝑚𝑞([𝑎1])𝑓)

′

𝐼𝑚𝑞([𝑎1])𝑓

}

> 𝛼 , where 𝛼 = (𝑚−_𝜆𝐴2_1(1−∣𝑎1∣𝐴∣_𝐴_∣)) which proves

the result. _□

Acknowledgments. The author would like to thank the anonymous referee for his/her valuable comments and suggestions to improve this article.

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Poonam Sharma

Department of Mathematics& Astronomy University of Lucknow Lucknow, 226007, UP INDIA