Applications of Differential Subordinations to Certain Classes of Starlike Functions

J. Korean Math. Soc. Published online September 24, 2019

SHAGUN BANGA and SIVAPRASAD KUMAR
Delhi Technological University

Abstract : Let p be an analytic function defined on the open unit disk D. We obtain certain differential subordination implications such as ᴪ(p)= p^(λ)(z)(α+β p(z) + ϒ/ p(z) + δ z p’(z)/p^(j) (z)) ‹ (subordinate) h(z) (j=1,2) implies p ‹ (subordinate) q, where h is given by ᴪ (q) and q belongs to carathѐodory class P, by finding the conditions on α, β, ϒ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy |log(z f'(z)/f(z))|<1, |(z f'(z)/f)^(2)-1|<1 and z f'(z)/f(z) lying in the parabolic region v^2 <2u-1.

Keywords : carathѐodory function, differential subordinations, minimum principle, exponential function, strongly starlike function, lemniscate of Bernoulli, janowski starlike function