• Subordination properties of certain integrals

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      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 > β, zεΔ, for some ⧎>0 and β< 1. Denote by S*(0)for B1(0,0). For μ andc such thatc > -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 > 0 so thatfεB1(μ -ρ) implies Iμ,c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB1 (μ -δ) impliesIμο(f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 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

       

      S Ponnusamy1

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

       
  • Proceedings – Mathematical Sciences | News

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