Class of bounds of the generalized Volterra functions

References

[1] Agarwal, R.—Kumar, N.—Parmar, R. K.—Purohit, S. D.: Some families of the general Mathieu-type series with associated properties and functional inequalities, Math. Methods Appl. Sci. 45 (2022), 2132–2150.10.1002/mma.7913Search in Google Scholar

[2] Ali, S.—Mubeen, S.—Ali, R. S.—Rahman, G.—Morsy, A.—Nisar, K. S.—Purohit, S. D.—Zakarya, M.: Dynamical significance of generalized fractional integral inequalities via convexity, AIMS Math. 6 (2021), 9705–9730.10.3934/math.2021565Search in Google Scholar

[3] Alzer, H.: On some inequalities for the incomplete gamma function, Math. Comp. 66 (1997), 771–778.10.1090/S0025-5718-97-00814-4Search in Google Scholar

[4] Apelblat, A.: Volterra Functions, Nova Science Publ. Inc., New York, 2008.Search in Google Scholar

[5] Apelblat, A.: Integral Transforms and Volterra Functions, Nova Science Publ. Inc., New York, 2010.Search in Google Scholar

[6] Apelblat, A.: Some integrals of gamma, polygamma and Volterra functions, IMA J. Appl. Math. 34 (1985), 173–186.10.1093/imamat/34.2.173Search in Google Scholar

[7] Barry, D. A.—Parlange, J.-Y.—Li, L.: Approximation of the exponential integral (Theis Well function), J. Hydrol. 227 (2000), 287–291.10.1016/S0022-1694(99)00184-5Search in Google Scholar

[8] Hopf, E.: Mathematical Problems of Radiative Equilibrium, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 31, Cambridge University Press, London, 1934.Search in Google Scholar

[9] Erdélyi, A.—Magnus, W.—Oberhettinger, F.—Tricomi, F. G.: Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.Search in Google Scholar

[10] Erdélyi, A.—Magnus, W.—Oberhettinger, F.—Tricomi, F. G.: Higher Transcendental Functions, McGraw-Hill, New York, (1955), 217–227.Search in Google Scholar

[11] Landau, E.: Über einige ältere Vermutungen und Behauptungen in der Primzahlentheorie, Math. Z. 1 (1918), 1–24.10.1007/BF01203613Search in Google Scholar

[12] Mathai, A. M.—Saxena, R. K.—Haubold, H. J.: The H-Functions: Theory and Applications, Springer, 2010.10.1007/978-1-4419-0916-9Search in Google Scholar

[13] Gautschi, W.: Some elementary inequalities relating to the gamma and incomplete gammafunction, J. Math. Phys. 38 (1959), 77–81.10.1002/sapm195938177Search in Google Scholar

[14] Gradshteyn, I. S.—Ryzhik, I. M.: Table of Integrals, Series, and Products, Corrected and Enlarged, Academic, New York, 1980.Search in Google Scholar

[15] Hardy, G. H.: Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Chelsea Publ. Co., New York, 1959.Search in Google Scholar

[16] El Kamel, J.—Mehrez, K.: A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions, Positivity 22 (2018), 1403–1417.10.1007/s11117-018-0584-3Search in Google Scholar

[17] Madland, D. G.—Nix, J. R.: New calculation of prompt fission neutron spectra and average prompt neutron multiplicities, Nucl. Sci. Eng. 81 (1982), 213–271.10.13182/NSE82-5Search in Google Scholar

[18] Mehrez, K.: Positivity of certain classes of functions related to the Fox H-functions with applications, Anal. Math. Phys. 11 (2021), Art. No. 114.10.1007/s13324-021-00553-wSearch in Google Scholar

[19] Mehrez, K.: Some geometric properties of a class of functions related to the Fox-Wright functions, Banach J. Math. Anal. 14 (2020), 1222–1240.10.1007/s43037-020-00059-wSearch in Google Scholar

[20] Mehrez, K.: New Integral representations for the Fox-Wright functions and its applications, J. Math. Anal. Appl. 468 (2018), 650–673.10.1016/j.jmaa.2018.08.053Search in Google Scholar

[21] Mehrez, K.—Sitnik, S. M.: Generalized Volterra functions, its integral representations and applications to the Mathieu-type series, Appl. Math. Comput. 347 (2019), 578–589.10.1016/j.amc.2018.11.004Search in Google Scholar

[22] Mehrez, K.—Sitnik, S. M.: Monotonicity properties and functional inequalities for the Volterra and incomplete Volterra functions, Integral Transforms Spec. Funct. 29 (2018), 875–892.10.1080/10652469.2018.1512107Search in Google Scholar

[23] Mitrinović, D. S.: Analytic Inequalities, Springer-Verlag, Berlin, 1970.10.1007/978-3-642-99970-3Search in Google Scholar

[24] Mitrinović, D. S.—Pecarić, J. E.—Fink, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.10.1007/978-94-017-1043-5Search in Google Scholar

[25] Parmar, R. K.—Agarwal, R.—Kumar, N.—Purohit, S. D.: Extended elliptic-type integrals with associated properties and Turán-type inequalities, Adv. Difference Equ. 2021 (2021), Art. No. 381.10.1186/s13662-021-03536-0Search in Google Scholar

[26] Pecarić, J. E.—Proschan, F.—Tong, Y. L.: Convex Functions, Partial Orderings, and Statistical Applications. Math. Sci. Eng., Vol. 187, Academic Press, 1992.Search in Google Scholar

[27] Selvakumaran, K. A.—Rajaguru, P.—Purohit, S. D.—Suthar, D. L: Certain geometric properties of the canonical Weierstrass product of an entire function associated with conic domains, J. Funct. Spaces 2022 (2022), Art. ID 2876673.10.1155/2022/2876673Search in Google Scholar

[28] Srivastava, H. M.—Mehrez, K.—Tomovski, Z.: New inequalities for some generalized Mathieu type series and the Riemann zeta function, J. Math. Inequal. 12 (2018), 163–174.10.7153/jmi-2018-12-13Search in Google Scholar

[29] Srivastava, H. M.—Kumar, A.—Das, S.—Mehrez, K.: Geometric properties of a certain class of Mittag–Leffler-type functions, Fractal Fract. 6 (2022), Art. No. 54.10.3390/fractalfract6020054Search in Google Scholar

[30] Touchard, J.: Sur la fonction gamma, Bull. Soc. Math. France 41 (1913), 234–242.10.24033/bsmf.930Search in Google Scholar

[31] Volterra, V.: Theoria delle potenze, dei logaritmi e delle funzioni di composizione, R. Acc. dei Lincei, Mem., ser. 5a. 11 (1916).Search in Google Scholar

[32] Widder, D. V.: The Laplace Transform, Princeton University Press, Princeton, 1946.Search in Google Scholar