COEFFICIENTS OF STRONGLY ALPHA-CONVEX AND STRONGLY GAMMASTARLIKE FUNCTIONS

D. K. THOMAS

《Tamkang Journal of Mathematics》 48卷1期 (2017/03) Pp. 17-29

https://doi.org/10.5556/j.tkjm.48.2017.2036

摘要

Let the function f be analytic in D = {z : |z| ＜ 1} and be given by f (z) = z + Σ_(n=2)^∞ a_n z^n. For 0 ＜ β ≤ 1, denote by C (β) and S^∗(β) the classes of strongly convex functions and strongly starlike functions respectively. For 0 ≤ α ≤ 1, 0 ＜ β ≤ 1 and 0 ≤ γ ≤ 1, let M(α, β) be the class of strongly alpha-convex functions defined by (The equation is abbreviated), and M^∗(γ, β) the class of strongly gamma starlike functions defined by (The equation is abbreviated). We give sharp bounds for the initial coefficients of f ∈M(α, β) and f ∈M^∗(γ, β), and for the initial coefficients of the inverse function f^(-1) of f ∈M(α, β) and f ∈M^∗(γ, β). These results generalise, improve and unify known coefficient inequalities for C (β) and S^∗(β).