An asymptotic formula for an integral in starlike function theory

Abstract:

The paper is concerned with the integral \[ H = \int _0^{2\pi }|f{|^\sigma }|F{|^\tau }{(\operatorname {Re} F)^\kappa }\;d\theta \] in which f is a function regular and starlike in the unit disc, $F = zf’/f$, and the parameters $\sigma ,\tau ,\kappa$ are real. A study of H is of interest since various well-known integrals in the theory, such as the length of $f(|z| = r)$, the area of $f(|z| \leqslant r)$, and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as $r \to 1$, is obtained for H when f is a starlike function of positive order $\alpha$, and the parameters satisfy $\alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0$. Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.