Sharp estimates on β are determined so that an analytic function p defined on the open unit disk in the complex plane normalized by p(0) = 1 is subordinate to some well known starlike functions with positive real part whenever 1 + βzp ′ (z), 1 + βzp ′ (z)/p(z), or 1 + βzp ′ (z)/p 2 (z) is subordinate to √ 1 + z. Our results provide sharp version of previously known results.2010 Mathematics Subject Classification. 30C45, 30C80.
A function f, analytic in the unit disc Δ, is said to be in the family Rn(α) if Re{(znf(z))(n+1)/(zn−1f(z))(n)}>(n+α)/(n+1) for some α(0≤α<1) and for all z in Δ, where n õ No, No={0,1,2,…}. The The class Rn(α) contains the starlike functions of order α for n≥0 and the convex functions of order α for n≥1. We study a class of integral operators defined on Rn(α). Finally an argument theorem is proved
In this paper, we investigate a generalized family of complex-valued harmonic functions that are multivalent, sense-preserving, and are associated with k-uniformly harmonic functions in the unit disk. The results obtained here include a number of known and new results as their special cases.
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