Monotonicity and Convexity Properties and Some Inequalities Involving $E_{n,p}(x)$

  • Ahmed Yakubu Department of Mathematics, Faculty of Physical Sciences, University for Development Studies, Nyankpala Campus, P. O. Box TL1350, Tamale, N/R, Ghana
  • Musah Sulemana Department of Mathematics, Faculty of Physical Sciences, University for Development Studies, Nyankpala Campus, P. O. Box TL1350, Tamale, N/R, Ghana
  • Iddrisu Mohammed Katali Department of Mathematics, Faculty of Physical Sciences, University for Development Studies, Nyankpala Campus, P. O. Box TL1350, Tamale, N/R, Ghana
Keywords: p-analogue of the exponential integral function, monotonicity, convexity

Abstract

In this paper, we established some monotonicity and convexity properties of the p-analogue of the exponential integral function. The increasing and decreasing, positive and negative, and convexity and concavity properties of the function were established and proved. Complete monotonicity of the function was also considered.

References

Bell, W. W. (2004). Special functions for scientists and engineers. Mineola, NY: Dover Publications.

Kotani, M., Amerniy, A., Ishiguro, E., & Kimura, T. (1955). Table of molecular integrals. Tokyo, Japan: Maruzen Co. Ltd.

Chiccoli, C., Lorenzutta, S., & Maino, G. (1989). Recent results for generalized exponential integrals. Computers in Mathematics and Applications, 19(5), 21-29. https://doi.org/10.1016/0898-1221(90)90098-5

Milgram, M. A. (1985). The generalized integro-exponential function. Mathematics of Computation, 44, 441-458. https://doi.org/10.1090/S0025-5718-1985-0777276-4

Nantomah, K., Merovci, F., & Nasiru, S. (2017). A generalization of the exponential integral and some associated inequalities. Honan Mathematical Journal, 39(1), 49-59. https://doi.org/10.5831/HMJ.2017.39.1.49

Salem, A. (2011). A q-analogue of the exponential integral. Afr. Mat., 24, 117-125. https://doi.org/10.1007/s13370-011-0046-6

Sroysang, B. (2014). The k-th derivative of the incomplete exponential integral function. Mathematica Aeterna, 2, 141-144.

Sulaiman, W. T. (2012). Turan inequalities of the exponential integral function. Communications in Optimization Theory, 1, 35-41.

Sroysang, B. (2013). Inequalities for the incomplete exponential integral function. Communications in Mathematical Applications, 4(2), 145-148.

Yakubu, A., Nantomah, K., & Iddrisu, M. M. (2020). A p-analogue of the exponential integral function and some properties. Advances in Inequalities and Applications, 2020(7), 1-9.

Yakubu, A., Nantomah, K., & Iddrisu, M. M. (2020). The i-th derivative of the p-analogue of the exponential integral function and some properties. Journal of Mathematical and Computational Science, 10(5), 1801-1807.

Yakubu, A., Abubakari, B., & Assan, F. K. (2023). Some properties and inequalities for a two-parameter generalization of the incomplete exponential integral function. Asian Research Journal of Mathematics, 19(10), 154-160. https://doi.org/10.9734/arjom/2023/v19i10737

Nantomah, K. (2017). Generalized Holder's and Minkowski's inequalities for Jackson's q-integral and some applications to the incomplete q-gamma function. Abstract and Applied Analysis, 6(2017), Article ID 9796873.

Kazarinoff, N. D. (1961). Analytic inequalities. New York: Holt, Rinehart and Winston.

Mitrinovic, D. S. (1970). Analytic inequalities. New York: Springer-Verlag.

Monica, M. A., & Nantomah, K. (2019). Some inequalities for the Chaudhry-Zubair Extension of the gamma function. Asian Research Journal of Mathematics, 14(1), 1-9. https://doi.org/10.9734/arjom/2019/v14i130117

Widder, D. V. (1946). The Laplace transform. Princeton Mathematical Series 6. Princeton University Press.

Widder, D. V. (1934). The inversion of the Laplace integral and the related moment problem. Transactions of the American Mathematical Society, 36, 107-200. https://doi.org/10.1090/S0002-9947-1934-1501737-7

Widder, D. V. (1931). Necessary and sufficient conditions for the representation of a function as a Laplace integral. Transactions of the American Mathematical Society, 33, 851-892. https://doi.org/10.1090/S0002-9947-1931-1501621-6

Maligranda, L. (2012). The AM-GM inequality is equivalent to the Bernoulli inequality. Math Intelligencer, 34(1), 2 pages. https://doi.org/10.1007/s00283-011-9266-8

Nantomah, K. (2018). A generalization of Nielsen's beta-function. International Journal of Open Problems in Computer Mathematics, 11(2), 16-26. https://doi.org/10.12816/0049058

Published
2023-12-16
How to Cite
Yakubu, A., Sulemana, M., & Katali, I. M. (2023). Monotonicity and Convexity Properties and Some Inequalities Involving $E_{n,p}(x)$. Earthline Journal of Mathematical Sciences, 14(2), 191-203. https://doi.org/10.34198/ejms.14224.191203
Section
Articles