Distribution of zeros of entire functions with a subharmonic majorant

Restrictions on the distribution of the zeros of entire functions $f\neq0$ on the complex plane are established under upper bounds ${\log |f|\leq M}$ by subharmonic functions $M$ on $\mathbb C$. These bounds make up a broad scale of inequalities for various characteristics of the distribution of the zeros of $f$ in terms of relevant characteristics of the of the Riesz mass distribution of the subharmonic function $M$. Various classes of generalized convex functions of the argument ($p$-trigonometrically convex functions) or modulus ($p$-power convex functions) of the complex variable are used as test objects in these integral inequalities. From the restrictions obtained uniqueness theorems are deduced, from which all results analogous to the case when no additional special restrictions are imposed on the distribution of the zeros can be derived. The results are sharp in the sense of their sensitivity to the addition or removal of a single zero. Subharmonic versions of these results are also obtained for functions on a disc. Bibliography: 39 titles.