Application of the Euler ’ s gamma function to a problem related to F . Carlson ’ s uniqueness theorem

In his work on F. Carlson’s uniqueness theorem for entire functions of exponential type, Q. I. Rahman [5] was led to consider an infinite integral and needed to determine the rate at which the integrand had to go to zero for the integral to converge. He had an estimate for it which he was content with, although it was not the best that could be done. In the present paper we find a result about the behaviour of the integrand at infinity, which is essentially best possible. Stirling’s formula for the Euler’s Gamma function plays an important role in its proof.

The proof of Theorem A is, in part, based on the following auxiliary result presented in [5] as Lemma 6.
Unless f is a constant, M (r) is an increasing function of r.This is all that was needed in the proof of Proposition 1, as given in [5].Here we prove a result (Theorem 1) from which it follows that Our approach to the problem is different.We relate it to Euler's Gamma function and make effective use of Stirling's formula to obtain the result.
In our proof of Theorem 1 we also use the fact that log M (r) is a convex function of log r and not simply a non-decreasing function r.Our proof clearly suggests how to prove that (1.2) is best possible, as far as the number 1/2 in r 2Q+1/2 goes.
2. Some facts about M (r) and the Stirling's formula.
Since we may write (2.1) in the form Hadamard's three-circles theorem may be expressed by saying that log M (r) is a convex function of log r.
In our case, M (r 2) it follows that if f (z) is a transcendental entire function, then there exists a number r 0 such that log M (r)/(log r) is an unbounded strictly increasing function of r for r ≥ r 0 .We know that log M (r) is continuous.In addition, it is a convex function of log r.It is known (see [2, p. 142]) that a continuous convex function has finite right-hand and left-hand derivatives at each point, and that these derivatives themselves are nondecreasing functions.

Stirling's formula.
Our proof of Theorem 1 uses Stirling's formula for the Gamma function defined by the Eulerian integral of the second kind Γ(z) = ∞ 0 e −t t z−1 dt whenever this integral converges (it being understood that t z−1 has its principal value), and defined by analytical continuation elsewhere.Stirling's formula says [4, p. 42] that where the power of z has its principal value and

Proof of Theorem 1.
Setting βr = u, we see that where g(z) := f (z/β) is an entire function.It is therefore enough to prove Theorem 1 in the special case where β = 1.Thus, we have to prove that if f (z) is an entire function such that Hereafter we shall simply write M (r) for M (r, f ) and M (s) for M (s, f ) because we see no confusion in doing so.
In view of all that has been said in § 2.1, for any s > 0 there is a constant and that there is an s 0 such that C(s) ≥ 1 for all s ≥ s 0 .Hence, for any s ≥ S 0 := max{s 0 , 10}, we have Taking (2.4) into account, it follows from (2.3) that and so Thus we see that Let us define In order to obtain a good upper estimate for max t≥1 ϕ(t) we note that ϕ (t) = 0 if and only if From this it is easily seen that ϕ (t) has one and only one zero in (1, ∞).
Using this fact about max t≥1 ϕ(t) in (3.3), we obtain the desired result. 2 Remark 1.The proof of Theorem 1 is of a somewhat wider scope than it might appear.In fact, the property of the function M (r) by which log M (r) is a convex function of log r is shared by some other functions associated with an entire function f .For example, if then, log M p (r) is a convex function of log r for any p > 0. This is a wellknown result of G. H. Hardy [3].If f (z) := ∞ n=0 a n z n , then for any r > 0, the maximum of |a n |r n for n ∈ {0, 1, 2, . ..} is called the maximum term.It is usually denoted by µ(r) and log µ(r) is known [7, pp. 30-31] to be a convex function of log r.Remark 2. Infinite integrals arise in various areas of pure and applied mathematics as well as in Statistics.They are also of interest to physicists and engineers.So, a result like Theorem 1 has the potential to be useful in the future.As an immediate application of the result, we state the following generalization of Theorem A where condition (1.1) has been replaced by a less restrictive one.
The proof of Corollary 1 requires only a minor modification in the proof of Theorem A, as given in [7], and so we omit it.

r
entire function and denote the maximum of |f (z)| on |z| = r by M (r).2Q M (r) e −πr dr < ∞, Q := L + L/2δ be a transcendental entire function.By considering F (z) := f (z) − f (0) + 1 if necessary, we may suppose that log M (r, f ) is a positive increasing convex function of log r in −∞ < log r < ∞.Note that (3.1) holds if and only if ∞ 0 r α M (r, F ) e −r dr < ∞ and that (3.2) holds if and only if r α+1/2 M (r, F ) e −r = O(1) as r → ∞.