Abstract
For analytic functions f and g; the growth of the Hadamard composition of their Gelfond–Leont’ev derivatives is investigated in terms of generalized orders. A relation between the behaviors of the maximal terms of the Hadamard composition of Gelfond–Leont’ev derivatives and those of the Gelfond–Leont’ev derivative of a Hadamard composition is established.
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References
J. Hadamard, “Theoreme sur le series entieres,” Acta Math., 22, 55-63 (1899).
J. Hadamard, “La serie de Taylor et son prolongement analitique. Scientia Phys.-Math., 12, 43-62 (1901).
L. Bieberbach, Analytische Fortsetzung, Springer, Berlin, 1955.
Yu. F. Korobeinik and N. N. Mavrodi, “Singular points of the Hadamard composition,” Ukr. Math. Zh., 42(12), 1711–1713 (1990).
M. K. Sen, “On some properties of an integral function f ∗ g,” Riv. Math. Univ. Parma (2), 8, 317–328 (1967).
M. K. Sen, “On the maximum term of a class of integral functions and its derivatives,” Ann. Pol. Math., 22, 291–298 (1970).
A. O. Gelfond and A. F. Leont’ev, “On a generalization of Fourier series,” Mat. Sbornik, 23(3), 477–500 (1957).
L. L. Luhova, O. M. Mulyava, and M. M. Sheremeta, “Properties of Hadamard composition for Gelfond–Leont’ev derivatives of analytic functions,” Ufa Math. J., 2(2), 90–101 (2010).
M. N. Sheremeta, “Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 100–108 (1967).
M. M. Sheremeta, Entire Dirichlet Series [in Ukrainian], ISDO, Kyiv, 1993.
G. Polya and G. Szego, Problems and Theorems in Analysis, Springer, New York, 1972.
J. Hadamard, “Essai sur l’etude des fonctions donnees par leur developpement de Taylor,” J. Math. Pures et Appl., 12, 154–186 (1892).
J. M. Whittaker, “The lower order of integral functions,” J. London Math. Soc., 8, 10–17 (1933).
M. M. Sheremeta, “On the connection between the growth of a function analytic in a disk and the moduli of coefficients of its Taylor series,” DAN USSR, No. 6, 729–731 (1966).
Yu. M. Gal’ and M. M. Sheremeta, “On the growth of analytic fuctions in a half-plane given by Dirichlet series,” DAN USSR, Ser. A., No. 12, 1964–1067 (1978).
Yu. M. Gal’, On the Growth of Analytic Functions Given by Dirichlet Series Absolutely Convergent in a Half-Plane [in Russian], manuscript deposited at VINITI, No. 4080-80, VINITI, Moscow, 1980.
M. Fujiwara, “On the relation between M(r) and the coefficients of a power series,” Proc. Imp. Acad. Japan., 8, 220–223 (1932).
L. S. Sons, “Regularity of growth and gap,” J. Math. Anal. Appl., 24, 296–306 (1968).
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 2, pp. 234–255 April–June, 2020.
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Sheremeta, M.M., Mulyava, O.M. Hadamard compositions of Gelfond–Leont’ev derivatives of analytic functions. J Math Sci 249, 769–785 (2020). https://doi.org/10.1007/s10958-020-04972-1
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DOI: https://doi.org/10.1007/s10958-020-04972-1