Representation of analytic functions in bounded convex domains on the complex plane

The paper is devoted to entire functions of exponential type and regular growth. Exceptional sets are investigated outside of which these functions have estimates from below that asymptotically coincide with their estimates from above. An explicit construction of an exceptional set, which consists of disks with centers at zeros of the entire function, is indicated. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Ya. Levin. It is proved that the zero set of an entire function is properly balanced if and only if each function analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is represented by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leont′ev on the representation of analytic functions in a convex domain to the case of a multiple zero set.