Abstract
In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.
Acknowledgement
The authors are thankful to the Deanship of Graduate Studies and Scientific Research at University of Bisha for supporting this work through the Fast-Tract Research Support Program.
-
Communicated by Tomasz Natkaniec
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.7913Suche 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.2021565Suche in Google Scholar
[3] Alzer, H.: On some inequalities for the incomplete gamma function, Math. Comp. 66 (1997), 771–778.10.1090/S0025-5718-97-00814-4Suche in Google Scholar
[4] Apelblat, A.: Volterra Functions, Nova Science Publ. Inc., New York, 2008.Suche in Google Scholar
[5] Apelblat, A.: Integral Transforms and Volterra Functions, Nova Science Publ. Inc., New York, 2010.Suche 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/34.2.173Suche 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)00184-5Suche 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.Suche in Google Scholar
[9] Erdélyi, A.—Magnus, W.—Oberhettinger, F.—Tricomi, F. G.: Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.Suche in Google Scholar
[10] Erdélyi, A.—Magnus, W.—Oberhettinger, F.—Tricomi, F. G.: Higher Transcendental Functions, McGraw-Hill, New York, (1955), 217–227.Suche in Google Scholar
[11] Landau, E.: Über einige ältere Vermutungen und Behauptungen in der Primzahlentheorie, Math. Z. 1 (1918), 1–24.10.1007/BF01203613Suche in Google Scholar
[12] Mathai, A. M.—Saxena, R. K.—Haubold, H. J.: The H-Functions: Theory and Applications, Springer, 2010.10.1007/978-1-4419-0916-9Suche 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/sapm195938177Suche in Google Scholar
[14] Gradshteyn, I. S.—Ryzhik, I. M.: Table of Integrals, Series, and Products, Corrected and Enlarged, Academic, New York, 1980.Suche in Google Scholar
[15] Hardy, G. H.: Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Chelsea Publ. Co., New York, 1959.Suche 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/s11117-018-0584-3Suche 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/NSE82-5Suche 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-wSuche 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-wSuche 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.2018.08.053Suche 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.2018.11.004Suche 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), 875–892.10.1080/10652469.2018.1512107Suche in Google Scholar
[23] Mitrinović, D. S.: Analytic Inequalities, Springer-Verlag, Berlin, 1970.10.1007/978-3-642-99970-3Suche in Google Scholar
[24] Mitrinović, D. S.—Pecarić, J. E.—Fink, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.10.1007/978-94-017-1043-5Suche 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/s13662-021-03536-0Suche 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.Suche 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 2876673.10.1155/2022/2876673Suche 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-2018-12-13Suche 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/fractalfract6020054Suche in Google Scholar
[30] Touchard, J.: Sur la fonction gamma, Bull. Soc. Math. France 41 (1913), 234–242.10.24033/bsmf.930Suche 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).Suche in Google Scholar
[32] Widder, D. V.: The Laplace Transform, Princeton University Press, Princeton, 1946.Suche in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Right algebras in Sup and the topological representation of semi-unital and semi-integral quantales, revisited
- Topological representation of some lattices
- Exact-m-majority terms
- Polynomials whose coefficients are generalized Leonardo numbers
- A study on error bounds for Newton-type inequalities in conformable fractional integrals
- Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
- Close-to-convex functions associated with a rational function
- Complete monotonicity for a ratio of finitely many gamma functions
- Class of bounds of the generalized Volterra functions
- Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
- Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line
- Solving Fredholm integro-differential equations involving integral condition: A new numerical method
- Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation
- Weighted 1MP and MP1 inverses for operators
- Non-commutative effect algebras, L-algebras, and local duality
- Operator Bohr-type inequalities
- Some results for weighted Bergman space operators via Berezin symbols
- Lower separation axioms in bitopogenous spaces
- Fisher information in order statistics and their concomitants for Cambanis bivariate distribution
- Irreducibility of strong size levels
Artikel in diesem Heft
- Right algebras in Sup and the topological representation of semi-unital and semi-integral quantales, revisited
- Topological representation of some lattices
- Exact-m-majority terms
- Polynomials whose coefficients are generalized Leonardo numbers
- A study on error bounds for Newton-type inequalities in conformable fractional integrals
- Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
- Close-to-convex functions associated with a rational function
- Complete monotonicity for a ratio of finitely many gamma functions
- Class of bounds of the generalized Volterra functions
- Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
- Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line
- Solving Fredholm integro-differential equations involving integral condition: A new numerical method
- Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation
- Weighted 1MP and MP1 inverses for operators
- Non-commutative effect algebras, L-algebras, and local duality
- Operator Bohr-type inequalities
- Some results for weighted Bergman space operators via Berezin symbols
- Lower separation axioms in bitopogenous spaces
- Fisher information in order statistics and their concomitants for Cambanis bivariate distribution
- Irreducibility of strong size levels