8. Certain New Classes of Analytic Functions Defined by Using the Salagean Operator

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 4(2010), Pages 62-75.

CERTAIN NEW CLASSES OF ANALYTIC FUNCTIONS DEFINED BY USING THE SALAGEAN OPERATOR

(DEDICATED IN OCCASION OF THE 70-YEARS OF PROFESSOR HARI M. SRIVASTAVA)

SHU-HAI LI, HUO TANG

Abstract. New classes of analytic functions deﬁned by using the Salagean operator are introduced and studied. We provide coeﬃcient inequalities, dis-tortion theorems, extreme points and radius of close-to-convexity, starlikeness and convexity of these classes.

1. _{INTRODUCTION AND DEFINITIONS}

Let𝒜denote the class of functions of the form

𝑓(𝑧) =𝑧+∑∞

𝑗=2𝑎𝑗𝑧𝑗, (1.1)

which are analytic in the open unit disc𝕌={𝑧:∣𝑧∣<1}.

Let𝒜+ _{denote the class of functions of the form}

𝑓(𝑧) =𝑧+∑∞

𝑗=2𝑎𝑗𝑧𝑗 (𝑎𝑗 ≥0), (1.2)

which are analytic in_𝕌.

We denote by 𝒮∗₍_{𝐴, 𝐵}_{) and} _𝒦₍_{𝐴, 𝐵}_{) (}₋₁ _≤ _{𝐵 < 𝐴} _≤ _{1) the subclasses of} starlike functions and the subclasses of convex functions, respectively, that is (see, for details, [1] and [2])

𝒮∗₍_{𝐴, 𝐵}_{) =}{_𝑓₍_𝑧₎_{∈ 𝒜}_: 𝑧𝑓′(𝑧) 𝑓(𝑧) ≺

1+𝐴𝑧

1+𝐵𝑧 (𝑧∈𝕌;−1≤𝐵 < 𝐴≤1) }

and

𝒦(𝐴, 𝐵) ={𝑓(𝑧)∈ 𝒜: 1 +𝑧𝑓_𝑓′′′₍(_𝑧𝑧₎)≺

1+𝐴𝑧

1+𝐵𝑧 (𝑧∈𝕌;−1≤𝐵 < 𝐴≤1) }

.

2000Mathematics Subject Classiﬁcation. Primary 30C45, 30C50; Secondary 26D15.

Key words and phrases. Analytic functions; Salagean operator; coeﬃcient inequalities; distor-tion inequalities; extreme points; radius.

c

⃝2010 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted March 19, 2010. Published November 12, 2010.

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Clearly, we have

𝑓(𝑧)∈ 𝒦(𝐴, 𝐵)⇐⇒𝑧𝑓′₍_𝑧₎_{∈ 𝒮}∗₍_{𝐴, 𝐵}₎_.

A function 𝑓(𝑧) ∈ 𝒜 is said to be in the class of uniformly convex functions, denoted by𝒰 𝒦 (see [3-5]) if

Re(1 +𝑧𝑓_𝑓′′′₍(_𝑧𝑧₎)

)

>

𝑧𝑓′′(𝑧) 𝑓′₍_𝑧₎

(1.3)

and is said to be in a corresponding class denoted by𝒰 𝒮 if

Re(𝑧𝑓_𝑓₍′_𝑧(𝑧₎))>

𝑧𝑓′(𝑧) 𝑓(𝑧) −1

. (1.4)

A function𝑓(𝑧)∈ 𝒜is said to be in the class of𝛼-uniformly convex functions of order𝛽, denoted by 𝒰 𝒦(𝛼, 𝛽) (see [6]) if

Re(1 +𝑧𝑓_𝑓′′′₍(_𝑧𝑧₎)

)

> 𝛼

𝑧𝑓′′(𝑧) 𝑓′₍_𝑧₎

+𝛽 (𝛼≥0; 0≤𝛽 <1) (1.5)

and is said to be in a corresponding class denoted by𝒰 𝒮(𝛼, 𝛽) if

Re(𝑧𝑓_𝑓₍′(_𝑧𝑧₎))> 𝛼

𝑧𝑓′(𝑧) 𝑓(𝑧) −1

+𝛽 (𝛼≥0; 0≤𝛽 <1). (1.6)

It is obvious that𝑓(𝑧)∈ 𝒰 𝒦(𝛼, 𝛽) if and only if𝑧𝑓′(𝑧)∈ 𝒰 𝒮(𝛼, 𝛽) (see [6]). The properties of various subclasses of functions 𝒰 𝒦(𝛼, 𝛽) and𝒰 𝒮(𝛼, 𝛽) were studied in [7].

For𝑓(𝑧)∈ 𝒜, Salagean [8] introduced the following operator which is called the Salagean operator:

𝐷0_𝑓₍_𝑧_{) =}_𝑓₍_𝑧₎_{, 𝐷}1_𝑓₍_𝑧_{) =}_𝑧𝑓′₍_𝑧₎_,_{⋅ ⋅ ⋅}_{, 𝐷}𝑛_𝑓₍_𝑧_{) =}_𝐷₍_𝐷𝑛−1_𝑓₍_𝑧_{)) (}_𝑛_∈_𝑁 ₌_{₁_,₂_,_{⋅ ⋅ ⋅ }}₎_.

We note that

𝐷𝑛𝑓(𝑧) =𝑧+∑∞ 𝑗=2𝑗

𝑛_𝑎

𝑗𝑧𝑗 (𝑛∈𝑁0=𝑁∪ {0}). (1.7)

Let 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) denote the subclass of 𝒜 consisting of functions 𝑓(𝑧) which

satisfy the following inequality:

𝐷𝑚_𝑓₍_𝑧₎

𝐷𝑛_𝑓₍_𝑧₎ −𝛼

𝐷𝑚_𝑓₍_𝑧₎

𝐷𝑛_𝑓₍_𝑧₎ −1

≺

1+𝐴𝑧

1+𝐵𝑧 (𝛼≥0,−1≤𝐵 < 𝐴≤1, 𝑚∈𝑁, 𝑛∈𝑁0).

(1.8) Also let𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵) (𝑠∈𝑁0) be the subclass of𝒜consisting of functions𝑓(𝑧)

which satisfy the following condition:

𝑓(𝑧)∈ 𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵)⇐⇒𝐷𝑠𝑓(𝑧)∈ 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵). (1.9)

For𝑠= 0, it is easy to see that

𝒱0

𝑚,𝑛(𝛼, 𝐴, 𝐵) =𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵).

When𝑚 = 1, 𝑛 = 0 and 𝑚= 2, 𝑛= 1 of inequality (1.8), respectively, we get two classes of functions

𝒰 𝒮(𝛼, 𝐴, 𝐵) ={𝑓(𝑧)∈ 𝒜: 𝑧𝑓_𝑓₍′(_𝑧𝑧₎)−𝛼

𝑧𝑓′(𝑧) 𝑓(𝑧) −1

≺

1+𝐴𝑧

1+𝐵𝑧, 𝛼≥0,−1≤𝐵 < 𝐴≤1 }

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𝒰 𝒦(𝛼, 𝐴, 𝐵) ={𝑓(𝑧)∈ 𝒜: 1 +𝑧𝑓_𝑓′′′₍(_𝑧𝑧₎)−𝛼

𝑧𝑓′′₍_𝑧₎

𝑓′₍_𝑧₎

≺

1+𝐴𝑧

1+𝐵𝑧, 𝛼≥0,−1≤𝐵 < 𝐴≤1 }

.

It is clear from two of the above deﬁnitions that

𝑓(𝑧)∈ 𝒰 𝒦(𝛼, 𝐴, 𝐵)⇐⇒𝑧𝑓′(𝑧)∈ 𝒰 𝒮(𝛼, 𝐴, 𝐵),

𝒰 𝒮(1,1,−1) =𝒰 𝒮, 𝒰 𝒦(1,1,−1) =𝒰 𝒦.

By specializing the parameters𝛼, 𝐴, 𝐵, 𝑚and𝑛involved in the class𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵),

we also obtain the following subclasses which were studied in many earlier works:

(1)𝒰1,0(𝛼,1−2𝛽,−1) =𝒰 𝒮(𝛼, 𝛽)𝑎𝑛𝑑𝒰2,1(𝛼,1−2𝛽,−1) =𝒰 𝒦(𝛼, 𝛽) (𝑠𝑒𝑒[6] ).

(2)𝒰𝑛+1,𝑛(𝛼,1−2𝛽,−1) =𝒰 𝒮𝑛(𝛼, 𝛽) (𝑠𝑒𝑒[9],[10] ).

(3)𝒰𝑚,𝑛(𝛼,1−2𝛽,−1) =𝒰𝑚,𝑛(𝛼, 𝛽)𝑎𝑛𝑑𝒱𝑚,𝑛𝑠 (𝛼,1−2𝛽,−1) =𝒱 𝑠

𝑚,𝑛(𝛼, 𝛽)

(0≤𝛼,0≤𝛽 <1)(𝑠𝑒𝑒[11],[12] ).

Let

˜

𝒰 𝒮(𝛼, 𝐴, 𝐵) =𝒜+_{∩ 𝒰 𝒮}₍_{𝛼, 𝐴, 𝐵}_{); ˜}_{𝒰 𝒦}₍_{𝛼, 𝐴, 𝐵}_{) =}_𝒜+_{∩ 𝒰 𝒦}₍_{𝛼, 𝐴, 𝐵}_);

˜

𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) =𝒜+∩ 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵); ˜𝒱𝑚,𝑛𝑠 (𝛼, 𝐴, 𝐵) =𝒜+∩ 𝒱𝑚,𝑛𝑠,+(𝛼, 𝐴, 𝐵).

Then we obtain contain relations and the close properties of integral operators. This paper mainly studies the classes𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) and𝒱𝑚,𝑛𝑠 (𝛼, 𝐴, 𝐵). We provide

coeﬃcient inequalities, distortion inequalities, extreme points and radius of close-to-convexity, starlikeness and convexity for the above classes.

2. _{COEFFICIENT INEQUALITIES FOR CLASSES} 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) AND

𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵)

Theorem 1. If𝑓(𝑧)∈ 𝒜satisﬁes

∑∞

𝑗=2𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)∣𝑎𝑗∣ ≤𝐴−𝐵 (2.1)

where

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = (1 + 2𝛼)∣𝑗𝑚₋_𝑗𝑛_∣₊_∣_𝐵𝑗𝑚₋_𝐴𝑗𝑛_∣ ₍₂_.₂₎

for some 𝛼 ≥ 0,−1 ≤ 𝐵 < 𝐴 ≤ 1, 𝑚 ∈ 𝑁, 𝑛 ∈ 𝑁0 = 𝑁 ∪ {0}, then 𝑓(𝑧) ∈ 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵).

Proof. Suppose that (2.1) is true for𝛼≥0,−1≤𝐵 < 𝐴≤1, 𝑚∈𝑁, 𝑛∈𝑁0.

For𝑓(𝑧)∈ 𝒜, let us deﬁne the function 𝑝(𝑧) by

𝑝(𝑧) = 𝐷

𝑚_𝑓₍_𝑧₎

𝐷𝑚𝑓(𝑧) 𝐷𝑛_𝑓₍_𝑧₎ −1

.

It suﬃces to show that

𝑝(𝑧)−1 𝐴−𝐵𝑝(𝑧)

<1 (𝑧∈𝕌).

We note that

𝑝(𝑧)−1 𝐴−𝐵𝑝(𝑧)

=

𝐷𝑚_𝑓₍_𝑧₎₋_𝛼𝑒𝑖𝜃_∣_𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎_{∣ −}_𝐷𝑛_𝑓₍_𝑧₎ 𝐴𝐷𝑛_𝑓₍_𝑧₎₋_𝐵₍_𝐷𝑚_𝑓₍_𝑧₎₋_𝛼𝑒𝑖𝜃_∣_𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎_∣₎

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=

(𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎₎₋_𝛼𝑒𝑖𝜃_∣_𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎_∣

(𝐴−𝐵)𝐷𝑛_𝑓₍_𝑧₎₋_𝐵₍₍_𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎₎₋_𝛼𝑒𝑖𝜃_∣_𝐷𝑚_𝑓₍_𝑧₎₋_𝐷𝑛_𝑓₍_𝑧₎_∣₎

=

∑∞ 𝑗=2(𝑗

𝑚₋_𝑗𝑛₎_𝑎

𝑗𝑧𝑗−1−𝛼𝑒𝑖𝜃∣∑∞_𝑗₌₂(𝑗𝑚−𝑗𝑛)𝑎𝑗𝑧𝑗−1∣

(𝐴−𝐵)−∑∞

𝑗=2(𝐵𝑗𝑚−𝐴𝑗𝑛)𝑎𝑗𝑧𝑗−1−𝛼𝑒𝑖𝜃∣∑∞𝑗=2(𝑗𝑚−𝑗𝑛)𝑎𝑗𝑧𝑗−1∣)

≤

∑∞ 𝑗=2∣𝑗

𝑚₋_𝑗𝑛_∣∣_𝑎

𝑗∣∣𝑧∣𝑗−1+𝛼∣𝑒∣𝑖𝜃∑

∞

𝑗=2∣𝑗

𝑚₋_𝑗𝑛_∣∣_𝑎 𝑗∣∣𝑧∣𝑗−1

(𝐴−𝐵)−∑∞

𝑗=2∣𝐵𝑗𝑚−𝐴𝑗𝑛∣∣𝑎𝑗∣∣𝑧∣𝑗−1−𝛼∣𝑒∣𝑖𝜃 ∑∞

𝑗=2∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣∣𝑧∣𝑗−1

≤

∑∞ 𝑗=2∣𝑗

𝑚₋_𝑗𝑛_∣∣_𝑎

𝑗∣+𝛼∑∞_𝑗₌₂∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣

(𝐴−𝐵)−∑∞

𝑗=2∣𝐵𝑗𝑚−𝐴𝑗𝑛∣∣𝑎𝑗∣ −𝛼∑∞𝑗=2∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣ .

The last expression is bounded above by 1, if

∞

∑

𝑗=2

∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣+𝛼

∞

∑

𝑗=2

∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣ ≤(𝐴−𝐵)−

∞

∑

𝑗=2

∣𝐵𝑗𝑚−𝐴𝑗𝑛∣∣𝑎𝑗∣−𝛼

∞

∑

𝑗=2

∣𝑗𝑚−𝑗𝑛∣∣𝑎𝑗∣

which is equivalent to the condition (2.1). This completes the proof of Theorem 1.

Corollary 1. If𝑓(𝑧)∈ 𝒜satisﬁes

∑∞

𝑗=2𝜙(1,0, 𝑗, 𝛼, 𝐴, 𝐵)∣𝑎𝑗∣ ≤𝐴−𝐵

where

𝜙(1,0, 𝑗, 𝛼, 𝐴, 𝐵) = (1 + 2𝛼)(𝑗−1) +∣𝐵𝑗−𝐴∣

for some𝛼≥0,−1≤𝐵 < 𝐴≤1, then𝑓(𝑧)∈ 𝒰 𝒮(𝛼, 𝐴, 𝐵).

Corollary 2. If𝑓(𝑧)∈ 𝒜satisﬁes

∑∞

𝑗=2𝜙(2,1, 𝑗, 𝛼, 𝐴, 𝐵)∣𝑎𝑗∣ ≤𝐴−𝐵

where

𝜙(2,1, 𝑗, 𝛼, 𝐴, 𝐵) = (1 + 2𝛼)𝑗(𝑗−1) +𝑗∣𝐵𝑗−𝐴∣

for some𝛼≥0,−1≤𝐵 < 𝐴≤1, then𝑓(𝑧)∈ 𝒰 𝒦(𝛼, 𝐴, 𝐵).

By using Theorem 1, we have

Theorem 2. If𝑓(𝑧)∈ 𝒜satisﬁes

∑∞ 𝑗=2𝑗

𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎_∣_𝑎

𝑗∣ ≤𝐴−𝐵

where𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is deﬁned by (2.2) for some𝛼≥0,−1≤𝐵 < 𝐴≤1, 𝑚∈

𝑁, 𝑛∈𝑁0, then 𝑓(𝑧)∈ 𝒱𝑚,𝑛𝑠 (𝛼, 𝐴, 𝐵).

Proof. From (1.7), Replacing𝑎𝑗 by𝑗𝑠𝑎𝑗 in Theorem 1, we have Theorem 2.

Example 1. The function𝑓(𝑧) given by

𝑓(𝑧) =𝑧+ ∞

∑

𝑗=2

(𝐴−𝐵)(2 +𝛿)𝜀𝑗

(𝑗+𝛿)(𝑗+ 1 +𝛿)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑧

𝑗 ₌_𝑧₊

∞

∑

𝑗=2

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with

𝐴𝑗 =

(𝐴−𝐵)(2 +𝛿)𝜀𝑗

(𝑗+𝛿)(𝑗+ 1 +𝛿)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

belongs to the class𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) for𝛿 >−2, 𝛼≥0,−1≤𝐵 < 𝐴≤1, 𝜀𝑗 ∈ℂand

∣𝜀𝑗∣= 1. Because, we know that

∞

∑

𝑗=2

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)∣𝐴𝑗∣ ≤

∞

∑

𝑗=2

(𝐴−𝐵)(2 +𝛿) (𝑗+𝛿)(𝑗+ 1 +𝛿)

= ∞

∑

𝑗=2

(𝐴−𝐵)(2 +𝛿) ∞

∑

𝑗=2

( 1 𝑗+𝛿−

1

𝑗+ 1 +𝛿) =𝐴−𝐵.

Example 2. The function𝑓(𝑧) given by

𝑓(𝑧) =𝑧+ ∞

∑

𝑗=2

(𝐴−𝐵)(2 +𝛿)𝜀𝑗

𝑗𝑠₍_𝑗₊_𝛿₎₍_𝑗_{+ 1 +}_𝛿₎_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎𝑧 𝑗₌_𝑧₊

∞

∑

𝑗=2

𝐵𝑗𝑧𝑗

with

𝐵𝑗=

(𝐴−𝐵)(2 +𝛿)𝜀𝑗

𝑗𝑠₍_𝑗₊_𝛿₎₍_𝑗_{+ 1 +}_𝛿₎_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎

belongs to the class𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵) for𝛿 >−2, 𝛼≥0,−1≤𝐵 < 𝐴≤1, 𝜀𝑗 ∈ℂand

∣𝜀𝑗∣= 1. Because, we know that

∞

∑

𝑗=2

𝑗𝑠𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)∣𝐵𝑗∣ ≤

∞

∑

𝑗=2

(𝐴−𝐵)(2 +𝛿)

(𝑗+𝛿)(𝑗+ 1 +𝛿)=𝐴−𝐵.

Theorem 3. If𝑓(𝑧)∈ 𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵), then for∣𝑧∣=𝑟 <1

1−(𝐴−𝐵)𝑟−𝐴𝐵𝑟2

1−𝐵2_𝑟2 ≤Re

{_𝐷𝑚_𝑓₍_𝑧₎

𝐷𝑚_𝑓₍_𝑧₎ 𝐷𝑛_𝑓₍_𝑧₎ −1

}

≤ 1 + (𝐴−𝐵)𝑟−𝐴𝐵𝑟 2

1−𝐵2_𝑟2 , 𝐵∕= 0, (2.4)

1−𝐴𝑟≤Re

{_𝐷𝑚_𝑓₍_𝑧₎

𝐷𝑚_𝑓₍_𝑧₎ 𝐷𝑛_𝑓₍_𝑧₎ −1

}

≤1 +𝐴𝑟, 𝐵= 0. (2.5)

Proof. Janowski [13] proved that if

𝑝(𝑧)≺ 1 +𝐴𝑧

1 +𝐵𝑧, ∣𝑧∣=𝑟 <1, then

𝑝(𝑧)−1−𝐴𝐵𝑟 2

1−𝐵2_𝑟2

<(𝐴−𝐵)𝑟

1−𝐵2_𝑟2, 𝐵∕= 0, (2.6)

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Using the deﬁnition of the class𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵), the inequality (2.6) and (2.7) can

be rewritten in the form

𝐷𝑚𝑓(𝑧) 𝐷𝑛_𝑓₍_𝑧₎ −𝛼 𝐷𝑚𝑓(𝑧) 𝐷𝑛_𝑓₍_𝑧₎ −1

−1−𝐴𝐵𝑟 2

1−𝐵2_𝑟2

< (𝐴−𝐵)𝑟

1−𝐵2_𝑟2, 𝐵∕= 0, (2.8)

𝐷𝑚𝑓(𝑧) 𝐷𝑛_𝑓₍_𝑧₎ −𝛼 𝐷𝑚𝑓(𝑧) 𝐷𝑛_𝑓₍_𝑧₎ −1

−1

< 𝐴𝑟, 𝐵= 0.(2.9)

From (2.8) and (2.9), we get (2.4) and (2.5) of Theorem 3.

Theorem 4 below follows easily from Theorem 3.

Theorem 4. If𝑓(𝑧)∈ 𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵), then for∣𝑧∣=𝑟 <1

1−(𝐴−𝐵)𝑟−𝐴𝐵𝑟2 1−𝐵2_𝑟2 ≤Re

{ 𝐷𝑚𝐷𝑠𝑓(𝑧) 𝐷𝑛_𝐷𝑠_𝑓₍_𝑧₎ −𝛼 𝐷𝑚𝐷𝑠𝑓(𝑧) 𝐷𝑛_𝐷𝑠_𝑓₍_𝑧₎ −1

}

≤1 + (𝐴−𝐵)𝑟−𝐴𝐵𝑟 2

1−𝐵2_𝑟2 , 𝐵∕= 0, (2.10)

1−𝐴𝑟≤Re

{_𝐷𝑚_𝐷𝑠_𝑓₍_𝑧₎ 𝐷𝑛_𝐷𝑠_𝑓₍_𝑧₎ −𝛼 𝐷𝑚_𝐷𝑠_𝑓₍_𝑧₎ 𝐷𝑛_𝐷𝑠_𝑓₍_𝑧₎ −1

}

≤1 +𝐴𝑟, 𝐵 = 0.(2.11)

Corollary 3. If𝑓(𝑧)∈ 𝒰 𝒮(𝛼, 𝐴, 𝐵), then for∣𝑧∣=𝑟 <1

1−𝐵2_𝑟2 ≤Re

{_𝑧𝑓′₍_𝑧₎ 𝑓(𝑧) −𝛼 𝑧𝑓′(𝑧) 𝑓(𝑧) −1

}

≤1 + (𝐴−𝐵)𝑟−𝐴𝐵𝑟 2

1−𝐵2_𝑟2 , 𝐵∕= 0,

1−𝐴𝑟≤Re

{_𝑧𝑓′₍_𝑧₎ 𝑓(𝑧) −𝛼 𝑧𝑓′(𝑧) 𝑓(𝑧) −1

}

≤1 +𝐴𝑟, 𝐵 = 0.

Corollary 4. If𝑓(𝑧)∈ 𝒰 𝒦(𝛼, 𝐴, 𝐵), then for∣𝑧∣=𝑟 <1

1−𝐵2_𝑟2 ≤Re {

1 + 𝑧𝑓 ′′₍_𝑧₎ 𝑓′₍_𝑧₎ −𝛼 𝑧𝑓′′(𝑧) 𝑓′₍_𝑧₎ }

≤1 + (𝐴−𝐵)𝑟−𝐴𝐵𝑟 2

1−𝐵2_𝑟2 , 𝐵∕= 0,

1−𝐴𝑟≤Re

{

1 +𝑧𝑓 ′′₍_𝑧₎ 𝑓′₍_𝑧₎ −𝛼 𝑧𝑓′′(𝑧) 𝑓′₍_𝑧₎ }

≤1 +𝐴𝑟, 𝐵= 0.

3. DISTORTION INEQUALITIES

Lemma 1. If𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵), then we have

∞

∑

𝑗=𝑝+1

𝑎𝑗≤

(𝐴−𝐵)−∑𝑝

𝑗=2𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗

𝜙(𝑚, 𝑛, 𝑝+ 1, 𝛼, 𝐴, 𝐵) , (3.1)

where𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is deﬁned by (2.2).

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∞

∑

𝑗=𝑝+1

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗 ≤(𝐴−𝐵)− 𝑝 ∑

𝑗=2

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗.(3.2)

Clearly,𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is an increasing function for 𝑗. Then from (2.2) and (3.2), we have

𝜙(𝑚, 𝑛, 𝑝+ 1, 𝛼, 𝐴, 𝐵) ∞

∑

𝑗=𝑝+1

𝑎𝑗≤(𝐴−𝐵)− 𝑝 ∑

𝑗=2

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗.

Thus, we obtain

∞

∑

𝑗=𝑝+1

𝑎𝑗 ≤

(𝐴−𝐵)−∑𝑝

𝑗=2𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗 𝜙(𝑚, 𝑛, 𝑝+ 1, 𝛼, 𝐴, 𝐵) =𝐴𝑗.

Lemma 2. If𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵), then

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗≤

(𝐴−𝐵)−∑𝑝

𝑗=2𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗

𝜙(𝑚−1, 𝑛−1, 𝑝+ 1, 𝛼, 𝐴, 𝐵) =𝐵𝑗, (3.3)

where𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is deﬁned by (2.2).

Corollary 5. If𝑓(𝑧)∈𝒱˜𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵), then

∞

∑

𝑗=𝑝+1

𝑎𝑗 ≤

(𝐴−𝐵)−∑𝑝 𝑗=2𝑗

𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎_𝑎 𝑗

(𝑝+ 1)𝑠_𝜙₍_{𝑚, 𝑛, 𝑝}_{+ 1}_{, 𝛼, 𝐴, 𝐵}₎ =𝐶𝑗 (3.4)

and

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗≤

(𝐴−𝐵)−∑𝑝

𝑗=2𝑗𝑠𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑎𝑗

(𝑝+ 1)𝑠_𝜙₍_𝑚₋₁_{, 𝑛}₋₁_{, 𝑝}_{+ 1}_{, 𝛼, 𝐴, 𝐵}₎ =𝐷𝑗.(3.5)

Theorem 5. Let𝑓(𝑧)∈_𝒰˜_𝑚,𝑛₍_{𝛼, 𝐴, 𝐵}_{). Then for}_∣_𝑧_∣₌_{𝑟 <}₁

𝑟− 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗−𝐴𝑗𝑟𝑝+1≤ ∣𝑓(𝑧)∣ ≤𝑟+ 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗+𝐴𝑗𝑟𝑝+1 (3.6)

and

1− 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1−𝐵𝑗𝑟𝑝≤ ∣𝑓′(𝑧)∣ ≤1 + 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1+𝐵𝑗𝑟𝑝 (3.7)

where𝐴𝑗 and𝐵𝑗 are given by Lemma 1 and Lemma 2.

Proof. Let𝑓(𝑧) given by (1.2). For∣𝑧∣=𝑟 <1, using Lemma 1, we have

∣𝑓(𝑧)∣ ≤ ∣𝑧∣+

𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗+

∞

∑

𝑗=𝑝+1

𝑎𝑗∣𝑧∣𝑗≤ ∣𝑧∣+ 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗+∣𝑧∣𝑝+1

∞

∑

𝑗=𝑝+1

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≤𝑟+

𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗+𝐴𝑗𝑟𝑝+1

and

∣𝑓(𝑧)∣ ≥ ∣𝑧∣ − 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗−

∞

∑

𝑗=𝑝+1

𝑎𝑗∣𝑧∣𝑗≥ ∣𝑧∣ − 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗− ∣𝑧∣𝑝+1

∞

∑

𝑗=𝑝+1

𝑎𝑗

≥𝑟− 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗−𝐴𝑗𝑟𝑝+1.

Furthermore, for∣𝑧∣=𝑟 <1, using Lemma 2, we also obtain

∣𝑓′(𝑧)∣ ≤1 +

𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1+

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗∣𝑧∣𝑗−1≤1 + 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1+∣𝑧∣𝑝

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗

≤1 +

𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1+𝐵𝑗𝑟𝑝

and

∣𝑓′(𝑧)∣ ≥1− 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1−

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗∣𝑧∣𝑗−1≥1− 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1− ∣𝑧∣𝑝

∞

∑

𝑗=𝑝+1

𝑗𝑎𝑗

≥1− 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1−𝐵𝑗𝑟𝑝.

This completes the assertion of Theorem 5.

Theorem 6. Let𝑓(𝑧)∈𝒱˜𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵). Then for∣𝑧∣=𝑟 <1

𝑟− 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗−𝐶𝑗𝑟𝑝+1≤ ∣𝑓(𝑧)∣ ≤𝑟+ 𝑝 ∑

𝑗=2

𝑎𝑗∣𝑧∣𝑗+𝐶𝑗𝑟𝑝+1 (3.8)

and

1− 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1−𝐷𝑗𝑟𝑝≤ ∣𝑓′(𝑧)∣ ≤1 + 𝑝 ∑

𝑗=2

𝑗𝑎𝑗∣𝑧∣𝑗−1+𝐷𝑗𝑟𝑝 (3.9)

where𝐶𝑗 and𝐷𝑗 are given by Corollary 5.

Proof. Using a similar method to that in the proof of Theorem 5 and making use Corollary 5, we get our result.

Taking𝑝= 1 in Theorem 5 and Theorem6, we have

Corollary 6. Let𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵). Then for∣𝑧∣=𝑟 <1

𝑟− 𝐴−𝐵

𝜙(𝑚, 𝑛,2, 𝛼, 𝐴, 𝐵)𝑟

2

≤ ∣𝑓(𝑧)∣ ≤𝑟+ 𝐴−𝐵 𝜙(𝑚, 𝑛,2, 𝛼, 𝐴, 𝐵)𝑟

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and

1− 2(𝐴−𝐵)

𝜙(𝑚, 𝑛,2, 𝛼, 𝐴, 𝐵)𝑟≤ ∣𝑓

′₍_𝑧₎_{∣ ≤}_{1 +} 2(𝐴−𝐵) 𝜙(𝑚, 𝑛,2, 𝛼, 𝐴, 𝐵)𝑟.

Corollary 7. Let𝑓(𝑧)∈𝒱˜𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵). Then for∣𝑧∣=𝑟 <1

𝑟− 𝐴−𝐵

2𝑠_𝜙₍_{𝑚, 𝑛,}₂_{, 𝛼, 𝐴, 𝐵}₎𝑟

2_{≤ ∣}_𝑓₍_𝑧₎_{∣ ≤}_𝑟₊ 𝐴−𝐵

2𝑠_𝜙₍_{𝑚, 𝑛,}₂_{, 𝛼, 𝐴, 𝐵}₎𝑟 2

and

1− 𝐴−𝐵

2𝑠−1_𝜙₍_{𝑚, 𝑛,}₂_{, 𝛼, 𝐴, 𝐵}₎𝑟≤ ∣𝑓

′₍_𝑧₎_{∣ ≤}_{1 +} 𝐴−𝐵

2𝑠−1_𝜙₍_{𝑚, 𝑛,}₂_{, 𝛼, 𝐴, 𝐵}₎𝑟.

Taking𝑝= 1, 𝑚= 1 and𝑛= 0 in Theorem 5 and Theorem 6, we also have

Corollary 8. Let𝑓(𝑧)∈𝒰 𝒮˜ (𝛼, 𝐴, 𝐵). Then for∣𝑧∣=𝑟 <1

𝑟− 𝐴−𝐵

𝜙(1,0,2, 𝛼, 𝐴, 𝐵)𝑟

2_{≤ ∣}_𝑓₍_𝑧₎_{∣ ≤}_𝑟₊ 𝐴−𝐵

𝜙(1,0,2, 𝛼, 𝐴, 𝐵)𝑟

2

and

1− 2(𝐴−𝐵)

𝜙(1,0,2, 𝛼, 𝐴, 𝐵)𝑟≤ ∣𝑓

′₍_𝑧₎_{∣ ≤}_{1 +} 2(𝐴−𝐵) 𝜙(1,0,2, 𝛼, 𝐴, 𝐵)𝑟.

Corollary 9. Let𝑓(𝑧)∈𝒰 𝒦˜ (𝛼, 𝐴, 𝐵). Then for∣𝑧∣=𝑟 <1

𝑟− 𝐴−𝐵

𝜙(2,1,2, 𝛼, 𝐴, 𝐵)𝑟

2_{≤ ∣}_𝑓₍_𝑧₎_{∣ ≤}_𝑟₊ 𝐴−𝐵

𝜙(2,1,2, 𝛼, 𝐴, 𝐵)𝑟

2

and

1− 2(𝐴−𝐵)

𝜙(2,1,2, 𝛼, 𝐴, 𝐵)𝑟≤ ∣𝑓

′₍_𝑧₎_{∣ ≤}_{1 +} 2(𝐴−𝐵) 𝜙(2,1,2, 𝛼, 𝐴, 𝐵)𝑟.

4. EXTREME POINTS

The determination of the extreme points of a family𝑓(𝑧) of univalent functions enables us to solve many extreme problems for𝑓(𝑧). Now, let us determine extreme points of the classes ˜𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) and ˜𝒱𝑚,𝑛𝑠 (𝛼, 𝐴, 𝐵).

Theorem 7. Let𝑓1(𝑧) =𝑧and

𝑓𝑗(𝑧) =𝑧+

𝐴−𝐵 𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝑧

𝑗 ₍_𝑗 _{= 2}_,₃_,_{⋅ ⋅ ⋅}₎_,

where𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is deﬁned by (2.2). Then𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵) if and only

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𝑓(𝑧) = ∞

∑

𝑗=1

𝜆𝑗𝑓𝑗(𝑧), (4.1)

where𝜆𝑗>0 and∑∞𝑗=1𝜆𝑗 = 1.

Proof. Suppose that

𝑓(𝑧) = ∞

∑

𝑗=1

𝜆𝑗𝑓𝑗(𝑧) =𝑧+

∞

∑

𝑗=1

𝜆𝑗

𝑗_.

Then ∞

∑

𝑗=2

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) 𝐴−𝐵

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)𝜆𝑗 = ∞

∑

𝑗=2

(𝐴−𝐵)𝜆𝑗 = (𝐴−𝐵)(1−𝜆1)< 𝐴−𝐵.

Thus,𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵) from the deﬁnition of the class of𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵).

Conversely, suppose that𝑓(𝑧)∈𝒰˜𝑚,𝑛(𝛼, 𝐴, 𝐵). Since

𝑎𝑗≤

𝐴−𝐵

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) (𝑗= 2,3,⋅ ⋅ ⋅), we may set

𝜆𝑗 =

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

𝐴−𝐵 𝑎𝑗 (𝑗= 2,3,⋅ ⋅ ⋅) and

𝜆1= 1−

∞

∑

𝑗=2

𝜆𝑗.

Then

𝑓(𝑧) = ∞

∑

𝑗=1

𝜆𝑗𝑓𝑗(𝑧).

This completes the proof of Theorem 7.

Corollary 10. Let𝑔1(𝑧) =𝑧and

𝑔𝑗(𝑧) =𝑧+

𝐴−𝐵

𝑗𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎𝑧

𝑗 ₍_𝑗_{= 2}_,₃_,_{⋅ ⋅ ⋅}₎_.

Then𝑔(𝑧)∈𝒱˜𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵) if and only if it can be expressed in the form

𝑔(𝑧) = ∞

∑

𝑗=1

𝜆𝑗𝑔𝑗(𝑧),

where𝜆𝑗>0 and∑∞𝑗=1𝜆𝑗 = 1.

Corollary 11. The extreme points of ˜𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵) are the functions𝑓1(𝑧) =𝑧

and

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Corollary 12. The extreme points of ˜𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵) are the functions𝑔1(𝑧) =𝑧

and

𝑔𝑗(𝑧) =𝑧+

𝐴−𝐵

𝑗𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎𝑧

𝑗 ₍_𝑗_{= 2}_,₃_,_{⋅ ⋅ ⋅}₎_.

Corollary 13. The extreme points of ˜𝒰 𝒮(𝛼, 𝐴, 𝐵) are the functions𝑓1(𝑧) =𝑧

and

𝐴−𝐵 𝜙(1,0, 𝑗, 𝛼, 𝐴, 𝐵)𝑧

𝑗 ₍_𝑗_{= 2}_,₃_,_{⋅ ⋅ ⋅}₎_.

Corollary 14. The extreme points of ˜𝒰 𝒦(𝛼, 𝐴, 𝐵) are the functions𝑓1(𝑧) =𝑧

and

𝐴−𝐵 𝜙(2,1, 𝑗, 𝛼, 𝐴, 𝐵)𝑧

𝑗 ₍_𝑗_{= 2}_,₃_, ⋅ ⋅ ⋅).

5. _{RADIUS OF CLOSE-TO-CONVEXITY, STARLIKENESS AND}

CONVEXITY

We concentrate upon getting the radius of close-to-convexity, starlikeness and convexity.

Theorem 8. Let the function𝑓(𝑧) deﬁned by (1.1) be in the class𝒰𝑚,𝑛(𝛼, 𝐴, 𝐵).

Then𝑓(𝑧) is close-to-convex of𝜇(0≤𝜇 <1) in∣𝑧∣< 𝑟𝜇(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜇(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜇)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) 𝑗(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥_{2) (5}_.₁₎

and𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) is deﬁned by (2.2).

Proof. We must show that∣𝑓′(𝑧)−1∣<1−𝜇for∣𝑧∣< 𝑟𝜇(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵). We

have

∣𝑓′(𝑧)−1∣ ≤

∞

∑

𝑗=2

𝑗∣𝑎𝑗∣∣𝑧∣𝑗−1.

Thus∣𝑓′(𝑧)−1∣<1−𝜇if

∞

∑

𝑗=2

( 𝑗

1−𝜇)∣𝑎𝑗∣∣𝑧∣ 𝑗−1

≤1.(5.2)

By Theorem 1, we have

∞

∑

𝑗=2

(1−𝜇)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

𝐴−𝐵 ∣𝑎𝑗∣ ≤1. (5.3)

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𝑗∣𝑧∣𝑗−1

1−𝜇 ≤

(1−𝜇)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) 𝑗(𝐴−𝐵)

Equivalently if

∣𝑧∣ ≤ {(1−𝜇)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

𝑗(𝐴−𝐵) }

1

𝑗−1 (𝑗 ≥2).(5.4)

The theorem follows from (5.4).

Then𝑓(𝑧) is starlike of 𝜂(0≤𝜂 <1) in∣𝑧∣< 𝑟𝜂(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜂(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜂)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) (𝑗−𝜂)(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥_{2) (5}_.₅₎

Proof. It suﬃces to show that∣𝑧𝑓_𝑓₍′_𝑧(𝑧₎)−1∣<1−𝜂 for∣𝑧∣< 𝑟𝜂(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵).

We have

∣𝑧𝑓

′₍_𝑧₎

𝑓(𝑧) −1∣ ≤

∑∞

𝑗=2(𝑗−1)∣𝑎𝑗∣∣𝑧∣𝑗−1

1−∑∞

𝑗=2∣𝑎𝑗∣∣𝑧∣𝑗−1 .

Thus∣𝑧𝑓_𝑓₍′_𝑧(𝑧₎)−1∣<1−𝜂 if

∞

∑

𝑗=2

(𝑗−1)∣𝑎𝑗∣∣𝑧∣𝑗−1

(1−𝜂) ≤1 (5.6)

By using (5.3), (5.6), we have

(𝑗−𝜂)∣𝑧∣𝑗−1

(1−𝜂) ≤

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) (𝐴−𝐵) or Equivalently

∣𝑧∣ ≤ {(1−𝜂)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

(𝑗−𝜂)(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥₂₎_.₍₅_.₇₎

Then𝑓(𝑧) is convex of𝜉(0≤𝜉 <1) in∣𝑧∣< 𝑟𝜉(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜉(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜉)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) 𝑗(𝑗−𝜉)(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥_{2) (5}_.₈₎

Proof. It suﬃces to show that

∣𝑧𝑓

′′₍_𝑧₎

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∣𝑧𝑓

′₍_𝑧₎

𝑓(𝑧) −1∣=∣

∑∞

𝑗=2𝑗(𝑗−1)𝑎𝑗𝑧𝑗−1

1 +∑∞

𝑗=2𝑗𝑎𝑗𝑧𝑗−1 ∣ ≤

∑∞

𝑗=2𝑗(𝑗−1)∣𝑎𝑗∣∣𝑧∣𝑗−1

1−∑∞

𝑗=2𝑗∣𝑎𝑗∣∣𝑧∣𝑗−1 .

The last expression above is bounded by (1−𝜉) if

∞

∑

𝑗=2

𝑗(𝑗−𝜉)∣𝑎𝑗∣∣𝑧∣𝑗−1

(1−𝜉) ≤1. (5.10)

In view of (5.9), it follows that (5.10) is true if

𝑗(𝑗−𝜉)∣𝑧∣𝑗−1

(1−𝜉) ≤

𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) (𝐴−𝐵) or Equivalently

∣𝑧∣ ≤ {(1−𝜉)𝜙(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵)

𝑗(𝑗−𝜉)(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥₂₎_.

And this completes the proof.

Corollary 15. Let the function𝑓(𝑧) deﬁned by (1.1) be in the class𝒱𝑠

𝑚,𝑛(𝛼, 𝐴, 𝐵).

Then𝑓(𝑧) is close-to-convex of𝜇(0≤𝜇 <1) in∣𝑧∣< 𝑟𝜇,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜇,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜇)𝑗𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎ 𝑗(𝐴−𝐵) }

1

𝑗−1 (𝑗≥2).

𝑚,𝑛(𝛼, 𝐴, 𝐵).

Then𝑓(𝑧) is starlike of 𝜂(0≤𝜂 <1) in∣𝑧∣< 𝑟𝜂,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜂,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜂)𝑗𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎

(𝑗−𝜂)(𝐴−𝐵) }

1

𝑗−1 ₍_𝑗≥₂₎_.

𝑚,𝑛(𝛼, 𝐴, 𝐵).

Then𝑓(𝑧) is convex of𝜉(0≤𝜉 <1) in∣𝑧∣< 𝑟𝜉,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) where

𝑟𝜉,𝑠(𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵) = inf 𝑗 {

(1−𝜉)𝑗𝑠_𝜙₍_{𝑚, 𝑛, 𝑗, 𝛼, 𝐴, 𝐵}₎ 𝑗(𝑗−𝜉)(𝐴−𝐵) }

1

𝑗−1 (𝑗 ≥2).

We remark in conclusion that, by suitably specializing the parameters involved in the results presented in this paper, we can deduce numerous further corollaries and consequences of each of these results.

Acknowledgments. The present investigation was partly supported by the Nat-ural Science Foundation of Inner Mongolia under Grant 2009MS0113 and Higher School Research Foundation of Inner Mongolia under Grant NJzy08150.

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[3] A.W.Goodman, On uniformly starlike functions, J. Math.Anal.Appl., 155(1991), 364-370. [4] W. C. Ma, Uniformly convex functions, Ann. Polon. Math., 57(1992), 165-175.

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[9] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math., 28(1997), 17-32. [10] Wei-Ping Kuang, Yong Sun and Zhi-Gang Wang, On Quasi-Hadamard product of certain classes of analytic functions, Bulletin of Mathematical Analysis and Applications, 1(2)(2009), 36-46.

[11] S. S. Eker, S. Owa, Certain classes of analytic functions involving Salagean operator, J. Inequal. Pure and Appl. Math., 10(1)(2009), 12-22.

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SHU-HAI LI

Department of Mathematics, Chifeng University, Chifeng, Inner Mongolia 024000, People’s Republic of China

E-mail address:[email protected]

HUO TANG

Department of Mathematics, Chifeng University, Chifeng, Inner Mongolia 024000, People’s Republic of China