Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (5): 1449-1456.doi: 10.1016/S0252-9602(10)60137-3

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ON A SUBCLASS OF CLOSE TO CONVEX FUNCTIONS

 PENG Zhi-Gang   

  1. Faculty of Mathematics and Computer Science, Hubei University, |Wuhan 430062, China
  • Received:2008-11-28 Revised:2009-04-14 Online:2010-09-20 Published:2010-09-20
  • Supported by:

    Supported partially by NSFC (10771053).

Abstract:

Let C'(α,β) be the class of functions f(z)=zn=2anzn analytic in D={z: |z|<1}, satisfying for some convex function g(z) with g(0)=g'(0)-1=0 and for all z in D the condition

|{zf'(z)/{g(z) -1/zf'(z)/{g(z)}+(1-2α)|<β

for some α,β(0≤α<1, 0<β≤1). A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C'(α, β). The results extend the work of C. Selvaraj.

Key words: starlike, convex, close to convex, radius of convexity

CLC Number: 

  • 30C45
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