On some problems in the theory of functions

The paper devoted the memory of the outstanding Russian mathematician A. F. Leont'ev (1917-1987) consists of three sections. In 1 the author sets one probably new hypothesis concerning the well known Riemann's Zeta Function ζ(z) and proves with the helps of this hypothesis that all zeros of ζ(z) are simple and lie only on the real axis and on the line Re z=1/2. In 2 the formulation of one theorem on convex functions from the first part of the monography of Hayman W. K. and Kennedy P. B. a bit is slightly corrected. In the last section the author express his gratitude to some mathematicians (especially to A. F. Leont'ev who supported him throughout his comparatively long scientific career.