Acta mathematica scientia, Series B >
ON A SUBCLASS OF CLOSE TO CONVEX FUNCTIONS
Received date: 2008-11-28
Revised date: 2009-04-14
Online published: 2010-09-20
Supported by
Supported partially by NSFC (10771053).
Let C'(α，β) be the class of functions f(z)=z+Σ∞n=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
PENG Zhi-Gang . ON A SUBCLASS OF CLOSE TO CONVEX FUNCTIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1449 -1456 . DOI: 10.1016/S0252-9602(10)60137-3
[1] MacGregor T H. Functions whose derivative has a positive real part. Trans Amer Math Soc, 1962, 104: 532--537
[2] Caplinger T R, Causey W M. A class of univalent functions. Proc Amer Math Soc, 1973, 39: 357--361
[3] Padmanabhan K S, On a certain class of functions whose derivatives have a positive real part in the unit disc. Ann Polon Math, 1970/1971, 23: 73--81
[4] Juneja O P, Mogra M L. A class of univalent functions. Bull Sci Math 2 S\'{e}rie, 1979, 103: 435--447
[5] Selvaraj C. A subclass of close-to-convex functions. Southeast Asian Bulletin of Mathematics, 2004, 28: 113--123
[6] Duren P. Univalent Functions. New York: Springer-Verlag, 1983
[7] Hallenbeck D J, MacGregor T H. Linear Problems and Convexity Techniques in Geometric Function Theory. Boston: Pitman Advanced Publishing Program, 1984
[8] Robertson M S. On the theory of univalent functions. Ann Math, 1936, 37: 374--408
[9] Janowski W. Some extremal problems for certain families of analytic functions. Ann Polon Math, 1973, 28: 297--326
[10] Goodman A W. Univalent Functions, Vol II. Tampa, FL: Mariner Publishing Co Inc, 1983
[11] Parvatham R, Shanmugham T N. On analytic functions with reference to an integral operator. Bull Austral Math Soc, 1983, 28: 207--215
[12] Aghalary R, Kulkarni S R. Some properties of the integral operators in univalent function. Studia Univ Babes-Bolyai Mathematica, 2001, 46(1): 3--9
/
| 〈 |
|
〉 |