On a certain subclass of strongly starlike functions

Abstract

<p>Let <em>S</em>*(α₁, α₂), where α₁, α₂ ∈ (0, 1], represent the class of functions <em>f </em>that are analytic in the open unit disk D, normalized by<em> f</em> (0) = <em>f ' </em>(0) − 1 = 0, and satisfying the following double-sided inequality:</p><p>−πα₁/ 2 < arg {<em>zf ' </em>(<em>z</em>) / <em>f </em>(<em>z</em>)} < πα₂ / 2 , (<em>z</em> ∈ D).</p><p>In this manuscript, we estimate the coefficients and logarithmic coefficients associated with functions that belong to the class <em>S*</em>(α₁, α₂). As a result, we provide a general bound for the coefficients of a strongly starlike function, which has been an open question until now. Finally, we derive upper and lower bounds for the expression Re{<em>zf</em> '(<em>z</em>) / <em>f</em> (<em>z</em>)}, where <em>f </em>∈ <em>S</em>*(α₁, α₂).</p>