Convolution properties of a class of starlike functions

Author:

Singh Ram,Singh Sukhjit

Abstract

Let R R denote the class of functions f ( z ) = z + a 2 z 2 + f(z) = z + {a_2}{z^2} + \cdots that are analytic in the unit disc E = { z : | z | > 1 } E = \{ z:\left | z \right | > 1\} and satisfy the condition Re ( f ( z ) + z f ( z ) ) > 0 , z E \operatorname {Re} (f’(z) + zf(z)) > 0,z \in E . It is known that R R is a subclass of S t {S_t} , the class of univalent starlike functions in E E . In the present paper, among other things, we prove (i) for every n 1 n \geq 1 , the n n th partial sum of f R , s n ( z , f ) f \in R,{s_n}(z,f) , is univalent in E E , (ii) R R is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of R R is a convex function in E E .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference8 articles.

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2. L. Fejěr, Uber die positivitat von summen, die nach trigonometrischen order Legendreschen funktionen fortschreiten, Acta Litt. Ac Sci. Szeged (1925), 75-86.

3. Functions starlike and convex of order 𝛼;Jack, I. S.;J. London Math. Soc. (2),1971

4. J. Krzyź, A counter example concerning univalent functions, Mat. Fiz. Chem. 2 (1962), 57-58.

5. An extremal problem for functions with positive real part;Robertson, M. S.;Michigan Math. J.,1964

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