Subordination properties of certain integrals

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, zeΔ, for some ⧎>0 and β -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 thatfeB1(μ -ρ) implies Iμ,c(f 0 so that feB1 (μ -δ) impliesIμο(f)eS*(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.