Certain Classes of Analytic Functions Involving Multiplier Transformations J. J. Bhamare

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8 IJLSM | May 2015, Available @ http://www.ijlsm.org

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Certain Classes of Analytic Functions Involving Multiplier Transformations

J. J. Bhamare Department of Mathematics,

S. S. V. P. S. B. S. Deore, College of Engineering, Deopur,Dhule-424002

M. S., India.

S. M. Khairnar

Department of Engineering Sciences,

MIT Academy of Engineering, Alandi, Pune-412105, M. S., India

Abstract- In this paper, we study certain subclasses of analytic functions involving multiplier transformations in the open unit disc. We derive few subordination results for the functions in these subclasses and discuss the applications of subordination results with the help of complex functions. We obtain coefficient estimates, radii of starlikeness, convexity and close-to-convexity, extreme points, and integral means inequalities, growth and distortion theorems for these classes.

Keywords: multiplier transformations, radii of convexity and close-to-convexity, extreme points, subordination etc.

AMS Subject Classification: 30C45 & 30C50

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I. INTRODUCTION AND PRELIMINARIES Let 𝒩 denote the class of functions of the form

𝑓 𝑧 ^𝛽 = 𝑧^𝛽+ 𝛽𝑎_𝑛𝑧^{𝛽+𝑛−1}

∞

𝑛=2

(1.1) which are analytic and univalent in the open disc 𝑈 = 𝑧 ∈ ∁: 𝑧 < 1 . Also consider the subclass ℳ of 𝒩 consisting of

functions of the form

𝑓 𝑧 ^𝛽 = 𝑧^𝛽 − 𝛽𝑎_𝑛𝑧^{𝛽+𝑛−1}

∞

𝑛=2

(𝛽 > 0 , 𝑎_𝑛≥ 0)

(1.2) For any integer m, we defined the multiplier transformations 𝐼_𝑚^𝑙 of functions 𝑓 ∈ 𝒩 by

𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 = 𝑧^𝛽 − 𝛽 𝛽 + 𝑙 𝛽 + 𝑙 + 𝑛 − 1

𝑚

𝑎_𝑛𝑧^{𝛽+𝑛−1}

∞

𝑛=2

= 𝑧^𝛽− 𝛽𝑄 𝑛, 𝛽, 𝑙 𝑎_𝑛𝑧^{𝛽+𝑛−1}, 𝑙 ≥ 0, 𝑧 ∈ 𝑈

∞

𝑛=2

(1.3) 𝑤𝑕𝑒𝑟𝑒 𝑄 𝑛, 𝛽, 𝑙 𝑎_𝑛 = ^{𝛽 +𝑙}

𝛽+𝑙+𝑛−1 𝑚

A function 𝑓 𝑧 ∈ ℳ is said to be in the class 𝑈𝑆𝑇(𝛼, 𝑘) (k-Uniformly starlike functions of order 𝛼) if it satisfies the condition:

𝑅𝑒 𝑧𝑓′ 𝑧 ^𝛽

𝑓 𝑧 ^𝛽 − 𝛼 > 𝑘 𝑧𝑓′ 𝑧 ^𝛽

𝑓 𝑧 ^𝛽 − 1 , 0 ≤ 𝛼 < 1, 𝑘 ≥ 0 𝑎𝑛𝑑 𝑧 ∈ 𝑈,

(1.4) And is said to be in the class 𝑈𝐶𝑉(𝛼, 𝑘) (k-Uniformly convex functions of order 𝛼) if it satisfies the condition:

𝑅𝑒 1 + 𝑧𝑓′′ 𝑧 ^𝛽

𝑓′ 𝑧 ^𝛽 − 𝛼 > 𝑘 𝑧𝑓′′ 𝑧 ^𝛽

𝑓′ 𝑧 ^𝛽 − 1 , 0 ≤ 𝛼 < 1, 𝑘 ≥ 0 𝑎𝑛𝑑 𝑧 ∈ 𝑈,

(1.5) Indeed it follows from (4) and (5) that

𝑓 ∈ 𝑈𝐶𝑉 𝛼, 𝑘 ⇔ 𝑧𝑓^′ ∈ 𝑈𝑆𝑇 𝛼, 𝑘

(1.6)

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The interesting geometric properties of these function classes were extensively studied by Waggas Galib Astan et. Al [16], Kanas et al. in [14, 15], and Murugusundaramoorthy and Magesh [3,4], Astan and Kulkarni[17].Astan and Buti[18], S. M. Khairnar and N. H. More [12], S. M. Khairnar and Meena More [10, 11].

Definition1.1: For two functions 𝑓 and 𝑔 analytic in U, we say that the function 𝑓 is subordinate to g in U, denoted by 𝑓 ≺ 𝑔, if their exist a Schwarz function 𝑤(𝑧), analytic in U with 𝑤 0 = 0 and 𝑤(𝑧) < 𝑧 < 1 (𝑧 ∈ 𝑈) , such that 𝑓 𝑧 = 𝑔 𝑤 𝑧 (𝑧 ∈ 𝑈).

In particular, if the function 𝑔 is univalent in U, the above subordination is equivalent to 𝑓 0 = 𝑔 0 , 𝑓 𝑈 = 𝑔 𝑈 . The Littlewood’s subordination theorem which we will use in our investigation to obtain the integral mean inequality.

Lemma1.2: If the functions 𝑓(𝑧) and 𝑔(𝑧) are analytic in U, with 𝑓(𝑧) ≺ 𝑔(𝑧), then 𝑓(𝑟𝑒^𝑖𝜃) ^𝜂𝑑𝜃 ≤

2𝜋

0

𝑔(𝑟𝑒^𝑖𝜃) ^𝜂𝑑𝜃

2𝜋

0

(1.7) Where 𝜂 > 0, 𝑧 = 𝑟𝑒^𝑖𝜃 𝑎𝑛𝑑 0 < 𝑟 < 1. Strict inequality holds for 0 < 𝑟 < 1

Unless 𝑓 is constant or 𝑤 𝑧 = 𝛼𝑧, 𝛼 = 1.

Let 𝜑(𝑧) be an analytic function in U with 𝜑 0 = 1, 𝜑^′ 0 > 0 𝑎𝑛𝑑 𝑅 𝜑 𝑧 > 0, 𝑧 ∈ 𝑈.

which maps the open unit disc U onto a region starlike with respect to 1 and is symmetric with respect to the real axis. Then, by𝑆^∗ 𝜑 𝑎𝑛𝑑 𝐶 𝜑 , respectively, denote the subclass of normalized analytic function class A which satisfy the following subordination relation

𝑧𝑓^′ 𝑧 ^𝛽

𝑓 𝑧 ^𝛽 ≺ 𝜑 𝑧 𝑎𝑛𝑑 1 + 𝑧𝑓^′′ 𝑧 ^𝛽

𝑓^′ 𝑧 ^𝛽 ≺ 𝜑 𝑧 , 𝑧 ∈ 𝑈.

These functions introduced and studied by Ma and Minda. For 𝜑 𝑧 =1 + 1 − 2𝛼 𝑧

1 − 𝑧 𝑧 ∈ 𝑈, 0 ≤ 𝛼 < 1 ;

These function classes reduce to the well known classes 𝑆^∗ 𝛼 of starlike functions of order 𝛼 in U and 𝐶 𝛼 of convex functions of order 𝛼 in U.

Definition1: For 𝛽 > 0, −1 ≤ 𝛼 < 1, 𝑙, 𝑘 ≥ 0 𝑎𝑛𝑑 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 1.3 , we difined a new class 𝒩_𝑚^𝑙(𝛼, 𝛽, 𝜆, 𝑘) as subclass of consisting function 𝑓 𝑧 ^𝛽 of the form (1.2) and satisfying analytic criterion

𝑅𝑒 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 𝛼 ≥ 𝑘 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 1 .

(1.8) II. COEFFICIENT ESTIMATE

We obtain Necessary and Sufficient condition for the function 𝑓 𝑧 ^𝛽 of the form (1.2) to belong to the class 𝒩_𝑚^𝑙(𝛼, 𝛽, 𝜆, 𝑘) Theorem2.1: A function 𝑓 𝑧 ^𝛽 is in the class 𝒩_𝑚^𝑙(𝛼, 𝛽, 𝜆, 𝑘)if and only if

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝛽𝑄 n, β, l 𝑎_𝑛

∞

𝑛=1

≤ 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

(2.1) The result is sharp for the function

𝑎_𝑛= 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝛽𝑄 n, β, l Where 𝛽 > 0, −1 ≤ 𝛼 < 1, 𝑘, 𝑙 ≥ 0, 0 ≤ 𝜆 < 1

𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 = 𝑧^𝛽− 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽+𝑛−1},

∞

𝑛=2

Proof: Suppose that (2.1) is true for 𝑧 ∈ 𝑈. Then 𝑅𝑒 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 𝛼 ≥ 𝑘 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 1

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𝑅𝑒 𝛽 − 𝛼 𝜆𝛽 + 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l β + n − 1 − 𝛼𝜆 β + n − 1 − 𝛼 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1 𝜆𝛽 + 1 − 𝜆 − ^∞_𝑛=1𝛽𝑄 n, β, l 𝜆 β + n − 1 + 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1

≥ 𝑘 𝛽 − 𝜆𝛽 + 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l β + n − 1 − 𝜆 β + n − 1 − 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1 𝜆𝛽 + 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l 𝜆 β + n − 1 + 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1 Allowing 𝑧 → 1 along with real axis, we get

1 + 𝑘 − 𝑘 + 𝛼 𝜆 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝛽𝑄 n, β, l 𝑎_𝑛 ≤

∞

𝑛=1

1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦, 𝑎𝑠𝑠𝑢𝑚𝑒 𝑡𝑕𝑎𝑡 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 , 𝑡𝑕𝑒𝑛

𝑘 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 1 − 𝑅𝑒 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 𝛼

≤ 1 + 𝑘 𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′

𝜆𝑧 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 ^′+ 1 − 𝜆 𝐼_𝑚^𝑙 𝑓 𝑧 ^𝛽 − 1

≤ 1 + 𝑘 𝛽 − 𝜆𝛽 − 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l β + n − 1 1 − 𝜆 − 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1 𝜆𝛽 + 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l β + n − 1 𝜆 + 1 − 𝜆 𝑎_𝑛𝑧^𝑛−1 is bounded above if ≤ 1 − 𝛼 , Letting 𝑧 → 1⁻𝑎𝑙𝑜𝑛𝑔 𝑡𝑕𝑒 𝑟𝑒𝑎𝑙 𝑎𝑥𝑖𝑠, 𝑤𝑒 𝑕𝑎𝑣𝑒

≤ 1 + 𝑘 𝛽 − 𝜆𝛽 − 1 − 𝜆 − ^∞_{𝑛 =1}𝛽𝑄 n, β, l β + n − 1 1 − 𝜆 − 1 − 𝜆 𝑎_𝑛 𝜆𝛽 + 1 − 𝜆 − ^∞_𝑛=1𝛽𝑄 n, β, l β + n − 1 𝜆 + 1 − 𝜆 𝑎_𝑛 β + n − 1 1 + 𝑘 − 𝜆 𝑘 + 𝛼 − 𝑘 + 𝛼 1 − 𝜆 𝛽𝑄 n, β, l 𝑎_𝑛

∞

𝑛=1

≤ 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 which completes the proof.

III. RADII OF STARLIKENESS, CONVEXITY AND CLOSE-TO-CONVEXITY

We obtain the radii of convexity and close-to- convexity results for 𝑓 𝑧 functions in the class𝒩_𝑚^𝑙(𝛼, 𝛽, 𝜆, 𝑘) in the following theorem

Theorem3.1: Let 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 . Then 𝑓 is starlike of order 𝛿, (0 ≤ 𝛿 < 1) in the disk 𝑧 < 𝑟 = 𝑟₁ 𝛼, 𝛽, 𝜆, 𝑘, 𝑛, 𝛿 , where

𝑟₁= 𝑖𝑛𝑓 𝑛 ≥ 2

𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.1) Proof: For 0 ≤ 𝛿 < 1, It is sufficient to show that

𝑓′ 𝑧 ^𝛽

𝑓 𝑧 ^𝛽 − 1 ≤ 1 − 𝛿 ,

(3.2) For 𝑧 ≤ 𝑟₁, we have

𝑍𝑓′ 𝑧 ^𝛽

𝑓 𝑧 ^𝛽 − 1 ≤ 𝛽 − 1 − ^∞_𝑛=2𝛽 𝛽 + 𝑛 − 2 𝑎_𝑛𝑧^𝑛−1

1 − ^∞_{𝑛 =2}𝛽𝑎_𝑛𝑧^𝑛−1 ≤ 1 − 𝛿

(3.3) Hence, (3.3) holds true if

𝛽 + 𝑛 − 𝛿 − 1 𝛽 𝛽 − 𝛿 𝑎_𝑛

∞

𝑛=2

𝑧 ^𝑛−1≤ 1

(3.4) With the aid of (2.1) and (3.4) is true if

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𝑧 ^𝑛−1≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

(3.5) Therefore,

𝑧 ≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.6) Setting 𝑧 = 𝑟₁ 𝛼, 𝛽, 𝜆, 𝑘, 𝑛, 𝛿 in (3.6), we get the radius of starlikeness, which completes the proof of theorem 3.1.

Theorem3.2: Let 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 . Then 𝑓 is convex of order 𝛿, (0 ≤ 𝛿 < 1) in the disk 𝑧 < 𝑟 = 𝑟₂ 𝛼, 𝛽, 𝜆, 𝑘, 𝑛, 𝛿 , where

𝑟₂= 𝑖𝑛𝑓 𝑛 ≥ 2

𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 1 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.7) Proof: For 0 ≤ 𝛿 < 1, it is sufficient to show

𝑧𝑓^′′ 𝑧 ^𝛽

𝑓^′ 𝑧 ^𝛽 ≤ 1 − 𝛿 ,

(3.8) For 𝑧 ≤ 𝑟₂, we have

𝑧𝑓^′′ 𝑧 ^𝛽

𝑓^′ 𝑧 ^𝛽 ≤ 𝛽 − 1 − ^∞_{𝑛 =2} β + n − 1 β + n − 2 𝑎_𝑛𝑧^𝑛−1

1 − ^∞_𝑛=2 β + n − 1 𝑎_𝑛𝑧^𝑛−1 ≤ 1 − 𝛿 ,

(3.9) Hence, (3.9) holds true if

𝛽 + 𝑛 − 1 𝛽 + 𝑛 − 𝛿 − 1 𝛽 − 𝛿

∞

𝑛=2

𝑎_𝑛 𝑧 ^𝑛−1≤ 1

𝑧 ^𝑛−1≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 1 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

(3.11) 𝑧 ≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙

𝛽 + 𝑛 − 1 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.12) Setting 𝑧 = 𝑟₂ 𝛼, 𝛽, 𝜆, 𝑘, 𝑛, 𝛿 in (3.14), we get the radius of convexity, which completes the proof of theorem 3.2.

Theorem3.3: Let 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 . Then 𝑓 is close to convex of order 𝛿, (0 ≤ 𝛿 < 1) in the disk 𝑧 < 𝑟 = 𝑟₃ 𝛼, 𝛽, 𝛾, 𝑛, 𝛿 , where

𝑟₃= 𝑖𝑛𝑓 𝑛 ≥ 2

𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 𝛿 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.13) Proof: For 0 ≤ 𝛿 < 1, it is sufficient to show that

𝑓′ 𝑧 ^𝛽

𝑧^𝛽−1 − 1 ≤ 1 − 𝛿

(3.14) For 𝑧 ≤ 𝑟₃, we have

𝑓′ 𝑧 ^𝛽

𝑧^{𝛽 −1} − 1 ≤ 𝛽 − 1 + 𝛽 𝛽 + 𝑛 − 1 𝑎_𝑛𝑧^𝑛−1

∞

𝑛=2

≤ 1 − 𝛿

(3.15)

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Hence, (3.15) holds true if

𝛽 𝛽 + 𝑛 − 1 𝛽 − 𝛿

∞

𝑛=2

𝑎_𝑛 𝑧 ^𝑛−1≤ 1

𝑧 ^𝑛−1 ≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

(3.17) Therefore,

𝑧 ≤ 𝛽 − 𝛿 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝛽 + 𝑛 − 1 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1𝑛−1

(3.16) Setting 𝑧 = 𝑟₃ 𝛼, 𝛽, 𝜆, 𝑘, 𝑛, 𝛿 in (3.21), we get the radius of close-to-convexity, which completes the proof of theorem 3.3.

IV. EXTREAM POINTS:

The extream points of the class 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 are given by the following theorem.

Theorem 4.1: Let 𝑓₁ 𝑧 ^𝛽 = 𝑧^𝛽,

𝑓_𝑛 𝑧 ^𝛽 = 𝑧^𝛽 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽 +𝑛−1}

(4.1) For n=2,3,4,…

Then, 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 if and only if it can be expressed in the form 𝑓 𝑧 ^𝛽 = 𝑌_𝑛

∞

𝑛 =1

𝑓_𝑛 𝑧 ^𝛽

(4.2) Where

𝑌_𝑛 ≥ 0 𝑎𝑛𝑑 𝑌_𝑛

∞

𝑛=1

=

(4.3) Proof: Suppose that f can be express as in (4.2). Our aim is to show that 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 .

By (4.2), we have that 𝑓 𝑧 ^𝛽 = 𝑌_𝑛

∞

𝑛=1

𝑓_𝑛 𝑧 ^𝛽 = 𝑌₁𝑓₁ 𝑧 ^𝛽+ 𝑌_𝑛

∞

𝑛=2

𝑓_𝑛 𝑧 ^𝛽

= 𝑌₁𝑓₁ 𝑧 ^𝛽

+ 𝑌_𝑛 𝑧^𝛽− 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽+𝑛−1}

∞

𝑛=2

(4.4) = 𝑌_𝑛𝑧^𝛽− 𝑌_𝑛𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽 +𝑛 −1}

∞

𝑛=1

= 𝑧^𝛽 − 𝛽𝑌_𝑛 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽 +𝑛−1}.

∞

𝑛=2

Now,

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1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

∞

𝑛=2

× 𝑌_𝑛 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽 +𝑛−1} = 𝑌_𝑛

∞

𝑛 =2

= 1 − 𝑌₁≤ 1

(4.5) Thus, 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 .

Conversely, assume that 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 . Since

𝑎_𝑛 ≤ 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑛 ≥ 2

(4.6) We can set

𝑌_𝑛= 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙

1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝑎_𝑛 𝑛 ≥ 2

(4.7) 𝑌₁= 1 − 𝑌_𝑛

∞

𝑛=2

Then,

𝑓 𝑧 ^𝛽 = 𝑧^𝛽− 𝛽

∞

𝑛 =2

𝑎_𝑛𝑧^{𝛽+𝑛−1}

= 𝑧^𝛽− 𝛽

∞

𝑛=2

𝑌_𝑛 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 𝑧^{𝛽 +𝑛 −1}

= 𝑧^𝛽− 𝑌_𝑛 𝑧^𝛽 − 𝑓_𝑛 𝑧 ^𝛽

∞

𝑛=2

(4.8)

= 𝑌₁𝑓₁ 𝑧 ^𝛽+ 𝑌_𝑛𝑓_𝑛 𝑧 ^𝛽

∞

𝑛=2

= 𝑌_𝑛𝑓_𝑛 𝑧 ^𝛽

∞

𝑛 =1

This completes the proof of Theorem4.1

V. INTEGRAL MEANS:

Theorem 5.1: Let 𝜂 > 0. If 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 𝑎𝑛𝑑 𝜑 𝛼, 𝛽, 𝑛 _𝑛=1^∞ are nondecreasing sequences, then, for 𝑧 = 𝑟𝑒^𝑖𝜃 𝑎𝑛𝑑 0 < 𝑟 <

1

𝑓(𝑟𝑒^𝑖𝜃)^𝛽 ^𝜂𝑑𝜃 ≤

2𝜋

0

𝑓₂(𝑟𝑒^𝑖𝜃)^𝛽 ^𝜂𝑑𝜃

2𝜋

0

(5.1)

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where

𝑓₂(𝑧)^𝛽 = 𝑧^𝛽 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧^{𝛽 +1}

(5.2) 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 𝑛 = 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙 .

Proof: Let 𝑓 of the form of (1.2) and

𝑓₂(𝑧)^𝛽 = 𝑧^𝛽 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧^{𝛽 +1}

(5.3) Then we must show that

1 − 𝛽𝑎_𝑛𝑧^𝑛−1

∞

𝑛=1

𝜂

𝑑𝜃 ≤

2𝜋

0

1 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧

𝜂

𝑑𝜃

2𝜋

0

(5.4) By lemma 5.2, it suffices to show that

1 − 𝛽𝑎_𝑛𝑧^𝑛−1

∞

𝑛=1

≺ 1 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧

(5.5) Setting

1 − 𝛽𝑎_𝑛𝑧^𝑛−1

∞

𝑛=1

= 1 − 𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑤(𝑧)

(5. 6) From (5.7) and (2.1) we obtain

𝑤(𝑧) = 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝛽 + 1 1 − 𝛾 + 1 − 𝛼

∞

𝑛=2

𝑎_𝑛𝑧^𝑛−1

≤ 𝑧 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2

1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

∞

𝑛=2

𝑎_𝑛

(5.7) ≤ 𝑧 < 1.

This completes the proof of Theorem 5.1.

VI. GROWTH AND DISTORSION THEOREM:

Theorem 6.1: If 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 , then,

𝑧 ^𝛽−𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆

𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧 ^{𝛽 +1}≤ 𝑓 𝑧 ^𝛽 ≤ 𝑧 ^𝛽 +𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 ,

(6.1) Where 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 𝑛 = 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙

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This results are sharp

Theorem 6.1: If 𝑓 ∈ 𝒩_𝑚^𝑙 𝛼, 𝛽, 𝜆, 𝑘 , then,

−𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝛽 + 1

𝛽 𝑧 ^𝛽−1𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧 ^𝛽 ≤ 𝑓^{′ 𝑧}^𝛽 ≤ 𝛽 𝑧 ^𝛽−1+𝛽 1 + 𝑘 𝛽 − 𝑘 + 𝛼 𝜆𝛽 + 1 − 𝜆 𝛽 + 1 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 2 𝑧 ^𝛽,

(6.2) Where 𝜑 𝛼, 𝛽, 𝜆, 𝑘, 𝑛 = 1 + 𝑘 − 𝜆 𝑘 + 𝛼 β + n − 1 − 𝑘 + 𝛼 1 − 𝜆 𝑄 𝑛, 𝛽, 𝑙

This results are sharp

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