On one application of Leontiev interpolation function in theory of trigonometrically convex functions

We study a connection between $\rho$ - trigonometrically convex functions and the class of subharmonic functions. The established connection is used to prove new inequalities characterizing $\rho$ - trigonometrically convex functions and to find integral equations of the first kind for $\rho$ - trigonometric functions. Under a detailed development of this issue, there appears the convolution integral equation $$ h(\theta)=\int\limits_{-\infty}^{\infty}h(\theta-u)d\sigma(u), $$ where $\sigma$ is a finite compactly supported measure. The results on the theory of this equation are exposed following A.F. Leontiev, who studied this equation in relation with the theory of Dirichlet series. Using the Leontiev interpolating function, we propose additional conditions ensuring that a continuous solution to the equation \begin{equation*} h(\theta)=\int\limits_{-\infty}^{\infty}a_R(u)h(\theta-u)du \end{equation*} for a fixed $R$ is a $\rho$ - trigonometric function.