Subordination properties of certain integrals
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Let B_{1}(μ,β) denote the class of functions f(z)= z + a_{2}z^{2}+ h+ a_{n}z^{m}+… 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 B_{1}(0,0). For μ andc such thatc > -μ, letF =I_{gm,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 thatfεB_{1}(μ -ρ) implies I_{μ},c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB_{1} (μ -δ) impliesI_{μο}(f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 0 in Δ then the nth partial sumf_{n} off satisfiesf_{n}(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.
S Ponnusamy^{1}
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Volume 129 | Issue 5
November 2019
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