Computational Methods and Function Theory 6 (2006), No. 1, 1--14
Copyright Heldermann Verlag 2006
Iossif V. Ostrovskii
iossif@fen.bilkent.edu.tr , Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey; Institute for Low Temperature Physics and Engineering, 47 Lenin ave, 61103, Kharkov, Ukraine.
In 1904, Hardy introduced an entire function depending on two parameters being a generalization of ez. He had studied in detail its asymptotic properties and that of its zeros. We consider the two following non-asymptotic problems related to the zeros. (i) Determine values of the parameters such that all the zeros belong to the open left half-plane. For these values, the analogs of sine and cosine generated by Hardy's function have real, simple and interlacing zeros. (ii) Determine the number of real zeros as a function of the parameters.
Keywords: Class P, integral representation, Levin's generalization of the Hermite-Biehler Theorem, logarithmic derivative, Rolle's Theorem.
MSC 2000: 30D10, 30D15.
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