Janowski introduced the class S*(A,B) and C(A,B) and obtain the growth and distortion theorem in the below paper:
"Janwoski, W.: Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28 (1973), 297-326."
C(A,B)={ f(z) is analytic: (1+zf''(z)/f'(z)) = (1+Aw(z)) / (1+Bw(z)) } where w(z) is an analytic in unit disk with w(0)=0, |w(z)| and -1 \leq B < A \leq 1.
This is a good question.
I suggest tackling this question in terms of the more general topic of coefficient bounds for classes of analytic functions. This is done in
S. Prema, B.S Keerthi, Coefficient bounds for certain subclasses of analytic functions, Journal of Math. Analysis 4, 2013, no. 1, 22-27:
http://olumcamp.ir/mathpapers/Perma%20and%20Keerthi,%20Coefficient%20Bounds%20for%20Certain%20Subclasses%20of%20Analytic%20Functions%20%282013%29.pdf
See Section 2, starting on page 23, for the coefficient bounds of the function class $P_{\sum}(\alpha,\lambda)$.
Another place to look is
H.M. Srivastava, D. Bansai, Coeffient estimates for a subclass of analytic and bi-univalent functions, J. of the Egyptian Math. Soc., 2014:
http://www.sciencedirect.com/science/article/pii/S1110256X14000509
Prof. James F Peters,
This is not a answer of my question. Your answer is related is related to Bi-univalent function. But my question is not related to that. So, please see again question.
Thanks.
Hi Navneet,
You can find the answer in the paper by Prof Melike Aydogan page 3 Theorem 4.
Yao Liang.
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