### A Study of Some Inequalities for Aleph-Function

**Shiv Shankar Sharma and Ashok Singh Shekhawat**

HOD, Department of Science and Humanities,Karpagam University, Coimbatore- 641021.TN, INDIA.

Department of Mathematics,Suresh Gyan Vihar University, Jaipur, Rajasthan, INDIA.

csmaths2004@yahoo.com,
toshankar506@gmail.com

(Received on: September 29, Accepted: October 5, 2017)

**ABSTRACT**

The aim of the present paper is to derive some inequalities for Alephfunction with the help of a number of inequalities for the gamma function by Nguyen Van Vinh1. The results obtained in the present paper are of manifold generality and basic in nature. On account of general nature of the results established, here a number of known and new results follow as their particular cases on suitable specifications of the parameters involved there in.

**Mathematics Subject Classification:**33C60, 26D20.

**Keywords:**heat transfer, mass transfer, MHD, chemical reaction, oscillating infinite vertical plate, variable temperature, porous medium.

**INTRODUCTION**

The Aleph (ℵ)-function, introduced by Südland2, however the notation and complete definition is presented here in the following manner in terms on Mellin-Barnes type integrals:

**FORMULATION OF THE PROBLEM**

The L = L_{iγ∞} is a suitable contour of the Mellin-Barnes type which runs from
γ - i∞ to γ + i∞ with γ∈R, the integers m, n, p_{i}, q_{i} satisfy the inequality 0 ≤ n_{1} ≤ p_{i},0 ≤ m_{1} ≤ q_{i},τ_{i} > 0; i = 1,..., r.The parameters A_{j}, B_{j}, A_{ji}, B_{ji} are positive real numbers and a_{j}, b_{j}, a_{ji}, b_{ji} are complex numbers such that the poles of ⌈(b_{j} + B_{j}ζ), j = 1,2,.......,m separating from those of ⌈(1 - a_{j} + A_{j}ζ) j = 1,2,.......,n All the poles of the integrand (1.1) are supposed to be easy and empty products are considered as unity. The existence conditions5 for
the Aleph-function2 are given below

For detail account of Aleph (ℵ)-function see^{9} and^{10}.

**Remark 1.** It is observed that there is no historical name given to (1.1), compared to^{2}. The Mellin transform of this function is the coefficient of z^{-ζ} in the integrand (1.1). There are no references containing table of ℵ-functions in the literature.

For τ_{1} = τ_{2} =...=τ_{r} =1 in (1.1) the definition of following I-function^{6} is discovered:

where Ω^{m1,n1}_{pi,qi,1;r} (ζ) is defined in (1.2). The existence conditions for the integral in (1.7) are the same as the given in (1.3) – (1.6) with τ_{1} = 1,i = 1,r.

If we further set r = 1, then (1.7) reduces to the familiar H-function given e.g. in the
monograph^{11}

where the kernel Ω^{m1,n1}_{pi,qi,1;1} (ζ) is given in (1.2), which itself is a generalization of Meijer’s G-function12 to which it reduces for A1=…=Ap = 1 = B1 =…= Bp.

**REQUIRED INEQUALITY**

The following inequality is required in our investigation due to^{1}:

**THE MAIN RESULTS**

We express the Aleph-function in terms of Mellin-Barnes type of contour integral by (1.2), we get

provided that

provided that

**Inequality III.** We put A_{i} = λ_{i}ζ + σ_{i} = p + 1 + λζ in (2.1), we get

multiplying all three terms with

and then integrating along the contour, we get

provided that

**Inequality IV.** We put A_{i} = σ_{i} -λ_{i}ζ, p - λζ in (2.1), we get

multiplying all three terms by

and then integrating along the contour, we get

provided that

**SPECIAL CASES**

- If we set τ
_{1}= τ_{2}=...=τ_{r}=1 in (3.1), (3.2), (3.2), (3.4), the Aleph-function occurring therein reduces to I-function and we arrive at the known results due to3. - If we set τ
_{1}= τ_{2}=...=τ_{r}=1 and r = 1 in inequalities (3.1), (3.2), (3.3), (3.4), we find the known result concluded by Ronghe, Kulwant and Arati4. - If we put τ
_{1}= τ_{2}=...=τ_{r}=1 , r = 1 and A_{j}B_{ji}A_{ji}B_{j}in the inequalities (3.1), (3.2), (3.3), (3.4), we find the known results reduces to Meijer’s G-function.

**CONCLUSION**

The Aleph-function, presented in this paper is quite basic nature. The Importance of the Aleph-function lies in the fact that the special functions I-function, H-function and Gfunction in the literature follow as its special cases. These special functions appear in various problems arising in theoretical and applied branches of mathematics, statistics, physics, engineering and others areas. The inequalities discussed in this paper can be useful for investigators in various disciplines of applied sciences and engineering.

**REFERENCES**

- Nguyen Van Vinh and An inequality for the gamma function. Ngo Phuoc Nauyen Ngoc,
*International Mathematical Forum*4 No.28, 1379-1382 (2009). - N. Südland, B. Baumann and T.F., Who knows about the Aleph-function?
*Fract. Calc. Appl. Anal*., 1(4), 401-402 (1998). - V.G. Gupta and N.K. Jangid, On Certain Inequality Pertaining to I-function.
*International Journal of Science and Research (IJSR)*, Vol.3, Issue 12, December (2014). - A.K. Ronghe, Kulwant Kaur and Arati Baurase,
*Double inequalities for the Fox’s H-function*, Vol.4, No.3, pp.313-318 (2012). - R.K. Saxena, T.K. Pogany, Mathieu-type series for the Aleph-function occurring in Fokker-Planck equation,
*Eur. J. Pure. Appl. Math*., 3(6), 958-979 (2010). - V.P. Saxena, The I-function, Ananaya Publishers, New Delhi (2008).
- C. Fox, The G and H-function as symmetrical Fourier kernels,
*Trans. Amer. Math. Soc*. 98, 395-429, (1961). - A.M. Mathai, R.K. Saxena, The H-function with applications in statistics and other discipline, Halshed Press, Wiley Eastern Limited, New Delhi, New York (1978).
- Rinku Jain and Kirti Arekar, Pathway integral operator associated with Aleph-function and General Polynomial,
*Global Journal of Science, Frontier Research Mathematics and Decision Sciences*, Vol.13 (2013). - E. R. Love, Some integral equations involving hypergeometric functions,
*Proc. Edin. Math. Soc*. 15(3), 169-198 (1967). - A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-function: Theory and Applications. Springer, New York, (2010).
- A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol.3, McGraw-Hill, New York, (2004).