DESCRIPTION OF SOME WEIGHTED EXPONENTIAL CLASSES OF SUBHARMONIC FUNCTIONS

The role of subharmonic functions in such sections of analysis as complex and real analysis is very significant. Such classes of functions are closely related to analytic harmonic functions and make an important contribution to the gen-eral theory of potential and mathematical physics. In the works of R. Nevanlinna and W. Heiman, parametric representations of subharmonic classes in the plane of functions, whose characteristic has a power growth at infinity, are obtained. The question of whether similar representations are true for weighted classes that admit a stronger growth at infinity (for example, the exponential growth) arises in the theory of entire and meromorphic functions. In this article, classes of subharmonic functions with Nevanlinna characteristic that is summable with exponential weight in a complex plane are introduced for consideration, and the representing measures of functions of such classes are studied. When proving the re-sults, methods of complex and functional analysis are used. An important role in the study is played by potentials based on the factors of the modified Weierstrass product. The proof of the main result is based on the use of auxiliary assertions formulated in the form of lemmas.

In complex and real analysis, potential theory and mathematical physics the value of subharmonic functions is very significant (see [1][2][3]5]). In the works R. Nevanlinna and W. Hayman (e.g., see [1]) the obtained definition of a class of subharmonic in the plane of the functions, the characteristics of which have exponential growth at infinity. The question of whether faithful same parametric representation for the weight classes, allowing for stronger growth at infinity, say exponential growth, occurs in the theory of entire and meromorphic functions (see [6]). This paper studied the representing measures of the functions of class , as well as the necessary and sufficient condition for such measures.

Statement of the main result
a is the integer part of a real number a .
( ) ( ) is a factor of the modified product by K. Weierstrass (see [7,8] The opposite is also true: let µ is some non-negative Borel measure in C , satisfying the condition (1), then it is possible to build explicitly subharmonic function of class , for which µ will be a representing measure.

Proof of auxiliary assertions
We require some auxiliary assertions for the proof of the theorem.
is a non-negative monotonically increasing function, for which where 0 (3) Proof. The convergence of the integral (2) implies that the We find the asymptotics of the last integral by applying the L'Hospital's rule.
( ) The lemma is proved.
Proof. It follows easily from the following simple arguments. It is clear that Then, ( The lemma is proved.

Lemma 3. Let u is an arbitrary subharmonic function in C , while it admits a representation in the form:
It is obvious that ( ) h z belongs to the class under consideration. We will show that We apply the estimate (see [1], p. 94): Let's get that: To continue the evaluation of the function, we can partition the complex plane into sets: Each of the rings k ∆ is divided into small rings and we use Lemma 2. Therefore, By Lemma 2, we obtain It is obvious that Having established the convergence of the last integral, we prove the Lemma.
To do this, imagine the inner integral as a sum: Let us prove the boundedness of the second integral by a constant, independently of r . Indeed, However, when We estimate the integral.
( ) ( ) That is This point is the maximum point, therefore
It is clear that in 0 1 α ≤ < the theorem is proved, since from the condition (5)