• S Ponnusamy

Articles written in Proceedings – Mathematical Sciences

• Differential subordinations concerning starlike functions

Denote byS* (⌕), (0≤⌕&lt;1), the family consisting of functionsf(z)=z+a2z2+...+anzn+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎&lt;1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)μ−1&lt;h(z) implies f∈S*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.

• Differential sobordination and Bazilevič functions

LetM(z)=zn+…,N(z)=zn+… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))&lt;0 implies Re(M(z)/N(z))&gt;0 in Δ whenever Re(λ(z))&gt;0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)&lt;1+z/1−z impliesp(z)&lt;1+z/1−z, for Reλ(z)&gt;0,z∈Δ.

In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)&lt;h(z) impliesp(z)&lt;ϕ(z)&lt;h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019