Zeros of operators on real entire functions of order less than two

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

APPLICATIONS

117, 81-83

(1986)

Zeros of Operators on Real Entire Functions of Order Less than Two C. L. PRATHER Department

of Mathematics,

Virginia Polytechnic Institute Blacksburg, Virginia 24061-4097 Submitted

and State

University,

by R. P. Boas

This note is motivated by ideas surrounding the following conjecture of Polya [S], which remains open: If the order of a real entire function f is less than two and f has only a finite number of complex zeros, then its derivatives, from a certain one onwards, will have no complex zeros at all. A real entire function is an entire function that is real valued on the real axis. Polya [6] has obtained the best result to date in the direction of the conjecture. He proves the result when the order of the real entire function is less than $ Furthermore, the number $ is the best that one can achieve by Polya’s method. The point to be made here is that stronger results along the lines of this conjecture are true when differentiation is replaced by more general differential operators. In fact, the result contained here is to some extent suggested by previous papers of the author [2, 8, 9, 10, 111, and the work of De Bruijn [3], the results of the latter constituting progress towards a proof of the Riemann hypothesis. In addition, the following proposition answers a question raised by Craven and Csordas [4, p. 2891. PROPOSITION. Supposef is a real entire function of order less than rtwo having its (possibly infinitely many) zeros in the strip Jim ZJ ,< A, A > 0. Suppose that

T(t) = fi

[ok exp(i;lt) + Wk exp( - ik)]

k=l

has only real zeros, where wk # 0 and 1> 0. Here G denotesthe complex conjugateofo.IfD=(d/dz)andL=AN“* then all the zeros of T(D) f(z) and thus all the zeros of (T(D))“f (z) are real, for m = 2, 3,..., where (T(D))‘==

(T(D))“-‘T(D). 81 0022-247X/86$3.00 Copynght 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

82

C. L. PRATHER

This result follows from a related theorem of De Bruijn [4, Theorem 91. Let the entire functionf(z) be of order less than two and suppose that f has a finite number of zeros outside the strip IIm(z)\ < Sz. If q(u) = CfZO A,uk, AN#O, has all its zeros on the unit circle (u\ = 1, then all but a finite number of the zeros of the function

satisfy IIm .zI < [Max(Q2

- N12, O)]“2.

Proof: An exponential change of variable, the condition i = 6Nm 1/2 and an application of De Bruijn’s result prove the proposition. We obtain similar results for more general operators. For example, if 2 > 0 (co~(At/N))~’ is a function having only real zeros. If f(z) is a real entire function having order less than two and has its zeros in /Im zl O, then all the zeros of (~os(ID/N))~2f(z) lie in IIm zl < (Max(A’ - 12, 0)} ‘j2. Now, taking A = AM- iI2 for fixed positive integer M, we get that [(~os(ID/N))~‘]~f(z) h as all its zeros on the real axis. Now, (cos(At/N))‘@ + exp( - t’n’/2) as N + co, uniformly for - CC< t < CC and I>O. If A= AAJ-“~ as before, then exp( -2-‘M-‘A2D2)f(z) has all its zeros inside (Im z( Q {Max(A’AZ/M, 0)) ‘I’, so that

has only real zeros. The most general operators L(D) for which L(D)f(z) has zeros in the strip Jim z(
fi

(l-w/A,)exp(w/A,,),

where A is a nonzero constant, a >/O, b and A, are real, and I,“=, A;* < cc. Functions of this form are called Laguerre-Polya functions. This result if given by Polya [7] and Benz [ 11. Nore added in proof: While the current note was in press, T. Craven, G. Csordas, and W. Smith informed the author that they have proved the conjecture of Polya as stated.

REFERENCES 1. E. BENZ, Uber lineare, verschiebungstreue Funktionaloperationen und die Nullstellen gamer Funktionen, Commenf. Math. Helv. 7 (1934), 243-289. 2. R. P. BOAS AND C. L. PRATHER, Final sets for operators on finite Fourier transforms, Houston

J. Math.

5 (1979),

29-36.

OPERATORS ON REAL FUNCTIONS

83

3. N. G. DE BRUIJN, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197-226. 4. T. CRAVEN AND G. CSORDAS, An inequality for the distribution of zeros of polynomials and entire functions, Pacific J. Math. 95 (1981), 263-280. 5. G. P(JLYA, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Sot. 49 (1943), 178-191.

6. G. P~LYA, uber die RealitLt der Nullstellen fast alter Ableitungen gewisser ganzer Funktionen, Math. Ann. 114 (1937), 622-634. 7. G. P~LYA, Sur les operations fonctionneles lineaires tchangeables avec la derivation et sur les zeros des polynomes, C. R. Acad. Sci. Paris. Ser. 1 Math. 183 (1926), 413414. 8. C. L. PRATHER,On some new and old theorems on final sets, Houston J. Math. 7 (1981) 407430.

C. L. PRATHER, Final sets for operators on classes of entire functions representable by a Fourier integral, J. Math. Anal. Appl. 82 (1981), 2W220. 10. C. L. PRATHER, Final sets for operators on real entire functions having order one, normal type, Proc. Amer. Math. Sot. 90 (1984), 363-369. 11. C. L. PRATHER, Zeros of operators on functions and their analytic character, Rocky 9.

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681-699.