Abstract

We consider the equation , where is a polynomial and is an entire function. Let be the zeros of a solution to that equation. Lower estimates for the products are derived. In particular, they give us a bound for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.

1. Introduction and Statement of the Main Result

In the present paper, we investigate the zeros of solutions to the initial problem where is a polynomial with complex in general coefficients and is an entire function. It is assumed that there are nonnegative constants , and an integer , such that The literature devoted to the zeros of solutions of homogeneous equations is very rich. Here the main tool is the Nevanlinna theory. An excellent exposition of the Nevanlinna theory and its applications to differential equations is given in book [1]. In connection with the recent results see interesting papers [2–13] (see also [14, 15]). At the same time the zeros of solutions to nonhomogeneous ODE were not enough investigated in the available literature. Here we can point out [16], only, in which the estimates for the sums of the zeros of solutions to (1) have been derived. In the present paper, lower estimates for the products of the zeros are obtained. In addition, we refine the main result from [16].

Enumerate the zeros of with their multiplicities in order of increasing absolute values: . Denote Now we are in a position to formulate the main result of the paper.

Theorem 1. Let condition (3) hold. Then the zeros of the solution to problem (1) satisfy the inequality where

The proof of this theorem is presented in the next two sections. Below we also suggest the sharper but more complicated bound for the products of the zeros.

2. Solution Estimates

Consider the equation where and are entire functions. Put .

Lemma 2. A solution of (7) satisfies the inequality

Proof. For a fixed and we have Integrating twice this equation in , we obtain Hence, where Due to the comparison lemma [17, Lemma III.2.1], we have , where is a solution of the equation Here is the Volterra operator defined by and, therefore, But for any positive nondecreasing we have Similarly, Thus from (15) it follows But This implies the required result.

Note that in our reasoning and can be arbitrary piecewise continuous functions.

Consider now (1). In this case In addition, since for any , we have Now Lemma 2 yields the following.

Corollary 3. A solution of problem (1) satisfies the inequality

This corollary is sharp: as it is well known a solution of the homogeneous equation is an entire function of order no more than ; see, for example, [1, Proposition 5.1]. Besides, our proof is absolutely different.

Corollary 4. Let condition (3) hold. Then a solution of problem (1) satisfies the inequality

3. Proof of Theorem 1

Lemma 5. Let an entire function satisfy the inequality with an integer Then its Taylor coefficients are subject to the inequalities

Proof. Let . Then the Taylor coefficients of satisfy the relation and . By the well-known inequality for the coefficients of a power series, Employing the usual method for finding extrema it is easy to see that the function takes its smallest value in the range for defined by Since , the lemma is proved.

The solution to (1) can be represented as , where is the solution to (1) with , , and ; is the solution to (1) with , , and ; is the solution to (1) with .

Corollary 4 impliesSince , we obtain Introduce the notations Denote by , , and the Taylor coefficients of , and , respectively. Then Lemma 5 yields Let be the Taylor coefficients of . Since , we have Let us consider the entire function Enumerate the zeros of with their multiplicities in order of nondecreasing absolute values and assume that

Lemma 6. Let be represented by (34) and let condition (35) hold. Then

This lemma is a particular case of Theorem 2.1 proved in [18].

Furthermore, consider the function , where is a solution to (1). Recall that . Due to (33), the Taylor coefficients of satisfy the inequalities and . Denote Clearly, So due to (37) Now Lemma 6 implies Since , we have proved the following result.

Lemma 7. Let condition (3) hold. Then where

Let us estimate . Recall that is defined by (31).

Lemma 8. One has , where and therefore .

Proof. Taking into account that we obtain Since , (40) implies . This and (43) prove the lemma.

Proof of Theorem 1. Since , from the previous lemma we get But , and therefore This and Lemma 7 prove the theorem.

4. Sums of Zeros and the Counting Function

In this section we derive a bound for sums of the zeros of solutions. To this end we need the following.

Theorem 9. Let be defined by (34) and let condition (35) hold. Then

This theorem is proved in [19] (see also [20, Section 5.1]). It gives us the inequality Since , we get Now (47) implies the following.