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Abstract In this paper, we investigate the Dirichlet problem associated with the α -translating equation. Using the Perron method and a family of grim reapers as barriers, we prove the existence of a solution on a strip of R 2 and the boundary data is formed by two copies of a convex function. [ABSTRACT FROM AUTHOR]

So far, the sharp bound of the expression | a 2 a 4 − a 3 2 | for the class C of close-to-convex functions has remained unknown. In this paper, we obtain the estimation of this expression, called the second Hankel determinant, for C 0 , i.e. the subset of C consisting of functions f that satisfy in the unit disk the inequality Re ( z f ′ ( z ) / g ( z ) ) > 0 with a starlike function g . Moreover, some remarks on the second Hankel determinant for the class S of univalent functions are made. It is proven that max ⁡ { | a 2 a 4 − a 3 2 | : f ∈ S } is greater than 1. [ABSTRACT FROM AUTHOR]

In this paper we will first give a positive answer to Kaiser's conjecture on ε -positive centers for convex curves and then present its two applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the order of starlikeness and strong starlikeness in the class of functions f ( z ) = z + a 2 z 2 + … analytic in | z | < 1 in the complex plane and satisfying Re { 1 + z f ″ ( z ) f ′ ( z ) } > | z f ′ ( z ) f ( z ) − 1 | , | z | < 1 . [ABSTRACT FROM AUTHOR]