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

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1 doi:.795/a A N N A L E S U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL. LXX, NO., 6 SECTIO A 75 8 M. A. QAZI Application of the Euler s gamma function to a problem related to F. Carlson s uniqueness theorem Abstract. 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.. Introduction... Carlson s theorem. Carlson s theorem says see [, Chapter 9]) that if f is an entire function such that fz) = Oe b z ) as z for some b < π and fn) = for n =, ±, ±,..., then fz) is identically zero. The example fz) := sin πz shows that here b = π is inadmissible. The following generalization of Carlson s theorem appears in [5] as Theorem 7. Theorem A. Let {λ n } be a sequence of real numbers such that n λ n L for all n Z and λ n λ m δ > for m n. Also, let fz) be an Mathematics Subject Classification. 3A5, 3A. Key words and phrases. Entire functions, Hadamard s three circles theorem, Euler s Gamma function.

2 76 M. A. Qazi entire function and denote the maximum of fz) on z = r by Mr). Suppose that.) r Q Mr) e πr dr <, Q := L + L/δ and that fλ n ) = for all n Z. Then fz). The proof of Theorem A is, in part, based on the following auxiliary result presented in [5] as Lemma 6. Proposition A. Let Mr) := max z =r fz), where f is an entire function and let Q be a positive number. Furthermore, let r Q Mr) e πr dr <. Then, r Q Mr) e πr as r. Unless f is a constant, Mr) is an increasing function of r. This is all that was needed in the proof of Proposition, as given in [5]. Here we prove a result Theorem ) from which it follows that.) r Q+/ Mr) e πr = O) as r. 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 we also use the fact that log Mr) is a convex function of log r and not simply a non-decreasing function r. Our proof clearly suggests how to prove that.) is best possible, as far as the number / in r Q+/ goes. Theorem. Let Mr) = Mr, f) := max z =r fz), where f is an entire function and suppose that r α Mr) e βr dr < for some α > and some β >. Then r α+/ Mr) e βr = O) as r.. Some facts about Mr) and the Stirling s formula... Convexity of log Mr) as a function of log r. Hadamard s threecircles theorem [6, p. 7] says: Let fz) be an analytic function, regular for r z r 3. Furthermore, let r < r < r 3, and let M, M, M 3 be the maxima of fz) on the three circles z = r, r, r 3, respectively. Then.) M logr 3/r ) M logr 3/r ) M logr /r ) 3. Since we may write.) in the form.) log Mr ) log r 3 log r log r 3 log r log Mr ) + log r log r log r 3 log r log Mr 3 ), Hadamard s three-circles theorem may be expressed by saying that log Mr) is a convex function of log r. In our case, Mr) := max z =r fz), where f is an entire function. Unless f is a constant, Mr) is a strictly increasing function of r. It is easily seen that fz) is a polynomial of degree n, that is fz) := n ν= a νz ν, a n if

3 Application of the Euler s gamma function and only if log Mr)/log r) n as r. From.) it follows that if fz) is a transcendental entire function, then there exists a number r such that log Mr)/log r) is an unbounded strictly increasing function of r for r r. We know that log Mr) is continuous. In addition, it is a convex function of log r. It is known see [, p. 4]) 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 uses Stirling s formula for the Gamma function defined by the Eulerian integral of the second kind Γz) = e t t z dt whenever this integral converges it being understood that t z has its principal value), and defined by analytical continuation elsewhere. Stirling s formula says [4, p. 4] that.3) Γz) = πz z / e z e Jz), where the power of z has its principal value and.4) < Jx) < / x) x > ). 3. Proof of Theorem. Setting βr = u, we see that r α Mr, f) e βr dr = β α+ u α Mu, g) e u du, where gz) := fz/β) is an entire function. It is therefore enough to prove Theorem in the special case where β =. Thus, we have to prove that if fz) is an entire function such that 3.) for some α >, then r α Mr, f) e r dr < 3.) r α+/ Mr, f) e r = O) as r. The result is trivial if fz) is a polynomial. So, let fz) be a transcendental entire function. By considering F z) := fz) f) + if necessary, we may suppose that log Mr, f) is a positive increasing convex function of log r in < log r <. Note that 3.) holds if and only if r α Mr, F ) e r dr < and that 3.) holds if and only if r α+/ Mr, F ) e r = O) as r. Hereafter we shall simply write Mr) for Mr, f) and Ms) for Ms, f) because we see no confusion in doing so.

4 78 M. A. Qazi In view of all that has been said in., for any s > there is a constant C = Cs) such that log Mr) log Ms) + C logr/s) r > ), and that there is an s such that Cs) for all s s. Hence, for any s S := max{s, }, we have A := r α Mr) e r dr Ms) = Ms) s C ΓC + α + ). Taking.4) into account, it follows from.3) that r α r/s) C e r dr ΓC + α + ) > π e C + α) C+α+/ e C+α), and so s α Ms) < A e e C+α s C+α π C + α) C+α+/. Thus we see that 3.3) s α Ms) < A e π { max e s/t) t t /}. t Let us define 3.4) ϕt) := e s/t) t t /. In order to obtain a good upper estimate for max t ϕt) we note that ϕ t) = if and only if Bt) := log s log t + ) =. t From this it is easily seen that ϕ t) has one and only one zero in, ). Let us call it τ s. We claim that s s /) τ s s s > S ).

5 Application of the Euler s gamma function For this, note that B s ) s s ) = log ) s /) s s s ) { } s = log + ) + ) s s s ) s > ) + ) s ) 4 s ) + s + s { s ) } s ) = 7 4 s ) s ) 3 ) } >, { s whereas B s ) s = log s /) s = log + ) s s <. It follows that if ϕt) is as in 3.4) then { max ϕt) = max t / e t s/t) t : s t s /) t s } since s / e s as s t / s /, e t e / e s and s/t) t e / as s. Using this fact about max t ϕt) in 3.3), we obtain the desired result. Remark. The proof of Theorem is of a somewhat wider scope than it might appear. In fact, the property of the function Mr) by which log Mr) is a convex function of log r is shared by some other functions associated with an entire function f. For example, if M p r) := π π /p fr e iθ ) dθ) p, p >, then, log M p r) is a convex function of log r for any p >. This is a wellknown result of G. H. Hardy [3]. If fz) := n= a nz n, then for any r >, the maximum of a n r n for n {,,,...} is called the maximum term. It is usually denoted by µr) and log µr) is known [7, pp. 3 3] to be a convex function of log r. Remark. 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 has the potential to be useful in the future. As an immediate application of the result, we state the following

6 8 M. A. Qazi generalization of Theorem A where condition.) has been replaced by a less restrictive one. Corollary. Let {λ n }, L and Q be as in Theorem A. Also, let fz) be an entire function and denote the maximum of fz) on z = r by Mr). Suppose that r Q γ Mr) e πr dr < for some γ < / and that fλ n ) = for all n Z. Then fz). The proof of Corollary requires only a minor modification in the proof of Theorem A, as given in [7], and so we omit it. References [] Boas, Jr., R. P., Entire Functions, Academic Press, New York, 954. [] Boas, Jr., R. P., A Primer of Real Functions, The Carus mathematical monographs, No. 3, The Mathematical Association of America, 96. [3] Hardy, G. H., The mean value of the modulus of an analytic function, Proc. London Math. Soc. 4 95), [4] Henrici, P., Applied and Computational Complex Analysis, Vol., A Wiley- Interscience publication), John Wiley & Sons, New York, 977. [5] Rahman, Q. I., Interpolation of Entire functions, Amer. J. Math ), [6] Titchmarsh, E. C., The Theory of Functions, nd ed. Oxford University Press, 939. [7] Valiron, G., Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 949. M. A. Qazi Department of Mathematics Tuskegee University Tuskegee, AL 3688 U.S.A. qazima@aol.com Received January 6, 6