• 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  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.

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