On location in a half-plane of zeros of perturbed first order entire functions

We consider the entire functions h(z) = ∞ ∑ k=0 akz k k! and h̃(z) = ∞ ∑ k=0 ãkz k k! (a0 = ã0 = 1; z, ak, ãk ∈ C, k = 1, 2, . . . ), provided ∞ ∑


Introduction and statement of the main result
Consider the entire functions (1) h(z) = ∞ k=0 a k z k k! andh(z) = ∞ k=0ã k z k k! (a 0 =ã 0 = 1; z, a k ,ã k ∈ C, k = 1, 2, . . . ) under the conditions Any function of the typeĥ with |b k | ≤ const c k 0 and c 0 ≥ 1, can be reduced to the form (1) with condition (2) if we take z = w/2c.
How small should be the quantity in order to provide the inequality (4) inf k=1,2,...
The literature on perturbations of the zeros of analytic functions is rather rich. In particular, the results obtained enable us to explore the relations between the zeros of the power series, their partial sums and tails, cf. [5], to estimate the distances between the zeros of entire functions and the zeros of their derivatives, [2,3,6,8]. The variation of the zeros of general analytic functions under perturbations was investigated, in particular, by P.
Rosenbloom [17]. He has established the perturbation result that provides the existence of a zero of a perturbed function in a given domain. In the case of entire functions the Rosenbloom's results have been refined in [9] (see also [11]). Of course we cannot survey the whole subject here and refer the reader to the just mentioned papers and books, and references given therein. However, to the best of our knowledge the above pointed problem was not not considered in the available literature although it is important, in particular, for localization of the zeros of perturbed functions. Our main tool is the recent norm estimates for solutions of the perturbed Lyapunov equation. Put is the Riemann zeta function. Below we show that condition (3) implies Finally, denote In Section 4 we suggest estimates for ψ(h), q and ξ(h, γ). Now we are in a position to formulate the main result of this paper. 4ξ(h, γ)(γ 2 q 2 + γq(1 + 2γψ(h))) < 1 be fulfilled. Then inequality (4) is valid.
The proof of this theorem is presented in the next section.
2. Proof of Theorem 1.1 Let C n be the complex n-dimension Euclidean space with a scalar product (., .) and the norm . = (., .). Denote by C n×n the set of n × n-matrices. For an A ∈ C n×n , λ k (A) (k = 1, . . . , n) are the eigenvalues taken with the multiplicities, σ(A) is the spectrum, r s (A) = max k |λ k (A)| is the spectral radius, A * is the adjoint one, and A is the spectral norm: A 2 = r s (A * A); I is the unit n × n-matrix.
For an integer n > 1, let us consider the polynomials As it is shown in [11] Lemma 5.2.1, p. 117, this matrix is similar to the following one

Location of zeros of entire functions
Put h n (z) = z n f n (1/z) = n k=0 a k z k k! andh n (z) = z nf n (1/z).
Due to Hurwitz theorem [15, p. 4] if z 0 is an m-fold zero of h(z), then every sufficient small neighborhood of z 0 contains m zeros counted with their multiplicities of each h n for all sufficiently large n. Thus from (3) for all sufficiently large n we have Lemma 2.1. Let condition (5) hold. Then the spectral radius r s (2γF n − I) of the matrix 2γF n − I satisfies the inequality This proves the lemma.
For an A ∈ C n×n assume that (6) r s (A) < 1 and put Hence, we easily have Lemma 2.2. Let A,Ã ∈ C n×n and condition (6) hold. If, in addition, Proof. Consider the discrete Lyapunov equation with given A ∈ C n×n , X ∈ C n×n should be found. It can be directly checked that Obviously, X ≤ χ(A). Thus the inequalities imply that X −Ã * XÃ is a positive definite operator and therefore by [7, Theorem 6.1] r s (Ã) < 1, as claimed.
Thus, for sufficiently large n, whereˆ n ≥ 0 andˆ n → 0 as n → ∞. In addition, Moreover, So, condition (8) is provided by the inequality By Lemma 2.3, for sufficiently large n we have Now letting n → ∞, we get the required result.

Perturbed polynomials
In this section we considerably simplify Theorem 1.1 in the case of the polynomials . . . , n). The theory of polynomials in spite its long history cf. [1,16] continues to attract an attention of many mathematicians, for example see [14,18,19]. However to the best of our knowledge the above pointed problem has not been considered even for polynomials.