• Subordination properties of certain integrals

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/101/03/0219-0226

• # Keywords

Differential subordination; univalent star-like; convex function

• # Abstract

Let B1(μ,β) denote the class of functions f(z)= z + a2z2+ h+ anzm+… that are analytic in the unit disc Δ and satisfy the condition Ref′(z)(f(z)/z)⧎-1 &gt; β, zεΔ, for some ⧎&gt;0 and β&lt; 1. Denote by S*(0)for B1(0,0). For μ andc such thatc &gt; -μ, letF =Igm,c(f) be defined by$$F(z) = \left[ {\frac{{\mu + c}}{{Z^c }}\int_0^z {f^\mu (t)} t^{c - 1} dt} \right]^{1/\mu } ,z \in \Delta .$$ The author considers the following two types of problems: (i) To find conditions on ⧎,c and ρ &gt; 0 so thatfεB1(μ -ρ) implies Iμ,c(f&lt;εS*(0); (ii) To determine the range of μ and δ &gt; 0 so that fεB1 (μ -δ) impliesIμο(f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) &gt; 0 in Δ then the nth partial sumfn off satisfiesfn(z)/z≺ -1 -(2/z)log(l -z)in Δ. Here, ≺ denotes the subordination of analytic functions with univalent analytic functions. As applications we also give few examples.

• # Author Affiliations

1. School of Mathematics, SPIC Science Foundation, 92, G N Chetty Road, Madras - 600017, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019