A Loewner approach to a coefficient inequality for bounded univalent functions

Abstract

The Loewner theory is used to obtain the sharp upper bound for the functional Re ⁡ { e 2 i θ ( a 3 − a 2 2 ) + 4 σ e i θ a 2 } \operatorname {Re} \{ {e^{2i\theta }}({a_3} - a_2^2) + 4\sigma {e^{i\theta }}{a_2}\} over the class of univalent functions f ( z ) = b ( z + a 2 z 2 + a 3 z 3 + … ) f(z) = b(z + {a_2}{z^2} + {a_3}{z^3} + \ldots ) which map the unit disc into itself; θ ∈ R , σ ∈ [ 0 , 1 ] \theta \in {\mathbf {R}},\sigma \in [0,1] and b ∈ ( 0 , 1 ] b \in (0,1] are fixed parameters.