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The paper is concerned with the integral  \[    H = ∫  0 2 π  | f | σ  | F | τ  ( Re ⁡  F ) κ  d θ  H = \int _0^{2\pi }|f{|^\sigma }|F{|^\tau }{(\operatorname {Re} F)^\kappa }\;d\theta     \]  in which  f  is a function regular and starlike in the unit disc,      F = z f ′ / f F = zf’/f      , and the parameters      σ  , τ  , κ  \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 ) f(|z| = r)      , the area of      f ( | z | ⩽  r ) 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 →  1 r \to 1      , is obtained for  H  when  f  is a starlike function of positive order      α  \alpha      , and the parameters satisfy      α  σ  + τ  + κ  &gt; 1 , τ  + κ  ⩾  0 , κ  ⩾  0 , σ  &gt; 0 \alpha \sigma + \tau + \kappa &gt; 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma &gt; 0      . Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.

An asymptotic formula for an integral in starlike function theory