### An Upper Bound for the Whittaker Constant W

Journal of the London Mathematical Society, Vol. 22 (1947), 305-311

Received 2 September, 1947; read 16 October, 1947.

Introduction

Let \(f(z)\) be an integral function whose maximum modulus satisfies
\[\overline{\lim_{r \rightarrow \infty}} \frac{\log M(r)}{r} \le 1,\]
and such that \(f(z)\) and all its derivatives have each at least one zero on or within the circle |*z*| = ρ. The Whittaker constant W is the lower bound of these numbers ρ for which at least one such \(f(z)\) exists not identically zero. Whittaker and Boas use equivalent definitions. Levinson and Boas proved that

0.7199 < W < 0.7399,

while Pondiczery conjectured that W = 2*e*^{–1} = 0.7357.... In this note I show that W < 0.7378, thus reducing the length of the interval in which W is known to lie by about 10 per cent.

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