On A Subclass Of N-Uniformly Multivalent Functions Used By Incomplete Beta Function

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Copyright to IJIRSET www.ijirset.com 3453

FUNCTIONS USED BY INCOMPLETE BETA

FUNCTION

Mr. Pravin Ganpat Jadhav

*Research Scholar, Shri Jagdishprasad Jhabarmal Tibrewala University, Jhunjhunu, Rajasthan

(India)

** Assistant Professor , Hon. B. J. Arts, commerce and Science College, Ale, Pune, India

ABSTRACT: In this paper we introduce the new subclasses and of n-uniformly convex and n-uniformly starlike functions which are analytic and multivalent with negative coefficients. The object of this paper is to study of the application of incomplete beta function to n-uniformly convex multivalent functions

and obtain several properties of the subclasses and .

Key words and phrases : Analytic functions , Multivalent functions , Coefficient estimate, Distortion theorem, Starlike functions , Convex functions, Close-to-close functions, n-Uniformly Convex functions, n-Uniformly Starlike functions, Radii of close-to-close convexity , starlikeness and convexity

I. INTRODUCTION

Let denote the class of functions of the form

………(1)

which are analytic and multivalent in the unit disk for

Let denote the subclass of consisting of functions of the form

……….(2)

Definition 1.1 : A function is said to be n-uniformly starlike of order and , denoted by if and only if

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Copyright to IJIRSET www.ijirset.com 3454 Definition 1.3 : A function is said to be n-uniformly close to convex of order

and if

The incomplete beta function is defined as

where and

which is motivated from Carlson Shaffer operator defined by

Definition 1.4 : A function is said to be in the class if it satisfies

Where

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Copyright to IJIRSET www.ijirset.com 3455 if and only if

PROOF: A function is in the class Therefore we have

………..(1.1)

Using the fact for ,

………..(1.2)

Letting

Therefore from (1.2) we have

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Copyright to IJIRSET www.ijirset.com 3456 =

From (1.3) we have

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Copyright to IJIRSET www.ijirset.com 3457 By mean value theorem we obtain

COROLLARY 1.1:- then

COROLLARY 1.2:- for we have

,

THEOREM 2. A function and defined by is in the class if and only if

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Let =

………..(2.1)

Using the fact for ,

………..(2.2)

Letting

Therefore from (1.2) we have

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Copyright to IJIRSET www.ijirset.com 3459 =

From (2.3) we have

The last inequality holds for all . letting yields

By mean value theorem we obtain

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,

III. GROWTH AND DISTORTION THEOREM

THEOREM 3:- If then

With equality hold for

PROOF :- Therefore from (1.1)

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Copyright to IJIRSET www.ijirset.com 3461 Similarly

Therefore

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Copyright to IJIRSET www.ijirset.com 3462 With equality hold for

Therefore

Similarly

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THEOREM 5:- If then

With equality hold for

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Copyright to IJIRSET www.ijirset.com 3464 Similarly

Therefore

THEOREM 6:- If then

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Therefore

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Copyright to IJIRSET www.ijirset.com 3466 Therefore

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

THEOREM 7:- If , then is close to convex of order in where

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Copyright to IJIRSET www.ijirset.com 3467 from (1.1)

That is

Observe that (7.1) is true if

Therefore

, which complete the proof.

THEOREM 8:- If , then is starlike of order in where

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We have

………(8.1)

Hence (8.1) holds true if

) ( )

Or equivalently

………..(8.2)

From (1.1) we have

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THEOREM 9:- If , then is convex of order in where

PROOF:- We know that is convex if and only if is starlike We must show that

Where

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Copyright to IJIRSET www.ijirset.com 3470 (9.1)

From (1.1) we have

Hence by using (9.1) and (9.2) we get

, which complete the proof.

V. CLOSURE THEOREM THEOREM 10 :

Let and for

Then if and only if can be expressed in the form

where and

PROOF: - Suppose can be expressed in the form

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Then

Therefore

Conversely, suppose that

We have

We take

and

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REFERENCES

[1] Jadhav P. G. and S. M. Khairnar, Certain subclasses of analytic and multivalent functions in the unit disc International J. of Math. Sci. & Engg. Appls. (IJMSEA) Vol. 6 No. VI (November, 2012), pp.175-192

[2] Jadhav P. G. and S. M. Khairnar , Certain family of meromorphic and multivalent functions defined by linear operator in the unit disc , International J. of Math. Sci. & Engg. Appls. (IJMSEA) Vol. 7 No. II (Mar, 2013), pp.15-33

[3] Jadhav P. G. and S. M. Khairnar , Some application of differential operator to analytic and multivalent function with negative coefficient , International J. of Manage.Studies, Statistics & App. Economics.(IJMSAE)ISSN 2250-0367, Vol. 3 No. I (June, 2013), pp.111-128. [4] S. M. Khairnar and Meena More, On a Class of Meromorphic Multivalent Functions with Negative Coefficients Defined by Ruscheweyh

Derivative, International Mathematical Forum, 3, 2008, no. 22, 1087 – 1097

[5] S. M. Khairnar and Meena More, On a subclass of multivalent -Uniformly starlike and convex functions defined by a linear operator ,

IAENG International Journal of Applied Mathematics , 39:3, IJAM_39_06 , 2009