Logarithmic image's convexity in the integral transforms theory

Background. The method of integral transforms is one of the most important analytical methods of mathematical modeling. Numerical methods and computational algorithms are developed on its basis. Image properties indirectly reflect the properties of the originals. Sometimes, for example, for Fourier images, these properties contain new information about the original. The article is devoted to the study of the logarithmic convexity of the image for a non-negative original.
Materials and methods. Methods of information geometry allowed us to establish the properties of integral Fourier transforms for the first time by studying the corresponding Fisher information matrix. Methods of Laplace, Mellin, Weierstrass and others integral transforms theory were also used in obtaining the results.
Results. Formula for the Fischer information matrix and stress tensor for randomized families of distributions associated with Laplace, Mellin, and Weierstrass integral transforms is found. The logarithmic image's convexity for a non-negative original is established. A new proof of the logarithmic convexity of the Gamma- function and the moments inequality of distribution is proposed.
Conclusions. The proposed methods can be useful in the study of special functions of mathematical physics, in the theory of fractional-order integrals. Having an explicit expression of the information matrix is important for statistical applications.