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A Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient Defined by Carlson and Shaffer …

International Mathematical Forum 01/2010; 5(37):1839-1848.

ABSTRACT The main objective of this paper is to obtain necessary and sufficient condition for a subclass of uniformly convex functions and correspond-ing subclass of starlike functions with fixed second coefficient defined by Carlson and Shaffer operator for the function f (z) in UCT (α, β). Furthermore, we obtain extreme points, distortion bounds and closure properties for f (z) in UCT (α, β) by fixing second coefficient.

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    ABSTRACT: A function f(z)=z+a 2 z 2 +⋯ is said to be uniformly convex in the unit disk U if (i) it is starlike in U, and (ii) for each circular arc γ in U with centre ζ in U, the arc f(γ) is convex; this class of functions was introduced by A. W. Goodman [J. Math. Anal. Appl. 155, No. 2, 364-370 (1991; Zbl 0726.30013)]. The author then defines a related class S p ={F:F starlike in U, F(z)=zf ' (z), f uniformly convex in U}. The author proves that a function f(z)=z+a 2 z 2 +⋯, analytic in U, belongs to S p if and only if |zf ' (z)/f(z)-1|<Rezf ' (z)/f(z) for all z∈U, looks at particular examples of functions in S p such as f(z)=z+z n /(2n-1), and proves that, if f is a normalised univalent function, then 1 rf(rz) belongs to S p if and only if r≤0·33217⋯. Finally, he proves that, if f(z)=z+a 2 z 2 +⋯ belongs to S p , then |a 2 |≤8/π 2 and |a n |≤8 (n-1)π 2 ∏ k=3 n (1+8 (k-2)π 2 ) for n≥3; also, that |f(z)/z|≤exp(14ζ(3)/π 2 ) (where ζ is the Riemann Zeta function), and that this inequality is best-possible.
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