Analogue of the Carleman's formula for $A(z)$-analytic functions

Nasridin M. Zhabborov^ab, Behzod E. Husenov<sup>b

a</sup> Tashkent State University of Economics, Tashkent, Uzbekistan
^b Bukhara State University, Bukhara, Uzbekistan

Abstract: In this paper, an analogue of the Carleman formula is proved for $A(z)$-analytic functions from the Hardy class. The idea of obtaining the Carleman formulas and the concept of the Carleman function for $A(z)$-analytic functions from the Hardy class belong to M.M. Lavrentiev. In the proof of Carleman's formula, $A(z)$-harmonic functions and the Poisson formula in lemniscates $L(a,r)$, compactly belonging to the domain under consideration $D\subset\mathbb{C}$, are used substantially.

Keywords: $A(z)$-analytic function, Hardy class, $A(z)$-lemniscate, multiple Cauchy integral formula for $A(z)$-analytic functions.

UDC: 517.54

Received: 02.01.2025
Received in revised form: 28.01.2025
Accepted: 25.03.2025

Language: English