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On the Neumann-Poincaré operator

Josef Král, Dagmar Medková (1998)

Czechoslovak Mathematical Journal

Let Γ be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L C 2 ( Γ ) the space of all complex-valued functions on Γ which are square integrable w.r. to the arc-length on Γ . Let L 2 ( Γ ) stand for the space of all real-valued functions in L C 2 ( Γ ) and put L 0 2 ( Γ ) = { h L 2 ( Γ ) Γ h ( ζ ) | d ζ | = 0 } . Since the Cauchy singular operator is bounded on L C 2 ( Γ ) , the Neumann-Poincaré operator C 1 Γ sending each h L 2 ( Γ ) into C 1 Γ h ( ζ 0 ) : = ( π i ) - 1 P . V . Γ h ( ζ ) ζ - ζ 0 d ζ , ζ 0 Γ , is bounded on L 2 ( Γ ) . We show that the inclusion C 1 Γ ( L 0 2 ( Γ ) ) L 0 2 ( Γ ) characterizes the circle in the class of all...

On the order of convolution consistence of the analytic functions with negative coefficients

Grigore S. Sălăgean, Adela Venter (2017)

Mathematica Bohemica

Making use of a modified Hadamard product, or convolution, of analytic functions with negative coefficients, combined with an integral operator, we study when a given analytic function is in a given class. Following an idea of U. Bednarz and J. Sokół, we define the order of convolution consistence of three classes of functions and determine a given analytic function for certain classes of analytic functions with negative coefficients.

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