59 research outputs found
Sufficient conditions for Janowski starlikeness
. These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike
Sufficient Conditions for Janowski Starlikeness
Let A,B,D,E∈[−1,1] and let p(z) be an analytic function defined on the open unit disk, p(0)=1. Conditions on A, B, D, and E are determined so that 1+βzp'(z) being subordinated to (1+Dz)/(1+Ez) implies that p(z) is subordinated to (1+Az)/(1+Bz). Similar results are obtained by considering the expressions 1+β(zp'(z)/p(z))
and 1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike
Sufficient Conditions for Starlike Functions Associated with the Lemniscate of Bernoulli
Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta
zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}.
Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or
p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies
p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that condition on \beta is
derived so that p(z)\prec(1+Az)/(1+Bz) when p(z)+\beta
zp'(z)/p(z)\prec\sqrt{1+z}. Few more problems of the similar flavor are also
considered
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