A note on extreme points of subordination classes

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Let s ( F ) s(F) denote the set of functions subordinate to a univalent function F F in Δ \Delta in the unit disc. Let B B denote the set of functions ϕ ( z ) \phi (z) analytic in Δ \Delta satisfying | ϕ ( x ) | > 1 |{\phi (x)}| > 1 and ϕ ( 0 ) = 0 \phi (0) = 0 . Let D = F ( Δ ) D = F(\Delta ) and λ ( w , ∂ D ) \lambda (w,\partial D) denote the distance between w w and ∂ D \partial D (boundary of D D ). We prove that if ϕ \phi is an extreme point of B B then ∫ 0 2 π log ⁡ λ ( F ( ϕ ( e i t ) ) , ∂ D ) d t = − ∞ \int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty } . As a corollary we prove that if F ∘ ϕ F \circ \phi is an extreme point of s ( F ) s(F) then ∫ 0 2 π log ⁡ λ ( F ( ϕ ( e i t ) ) , ∂ D ) d t = − ∞ \int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty } .