Refinements of Some Recent Inequalities for Certain Special Functions

Abstract The aim of this paper is to give some refinements to several inequalities, recently etablished, by P.K. Bhandari and S.K. Bissu in [Inequalities via Hölder’s inequality, Scholars Journal of Research in Mathematics and Computer Science, 2 (2018), no. 2, 124–129] for the incomplete gamma function, Polygamma functions, Exponential integral function, Abramowitz function, Hurwitz-Lerch zeta function and for the normalizing constant of the generalized inverse Gaussian distribution and the Remainder of the Binet’s first formula for ln Γ(x).


Introduction
Throughout this section, p, q are conjugate exponents, that is p, q > 1 and 1 p + 1 q = 1. K is the real or complex field. For any continuous functions u, v : [a, b] → K, we recall the integral version of the Hölder's inequality: Modifying an inequality of C. Mortici ([8]), P.K. Bhandari and S.K. Bissu in [3] replaced u(t) and v(t) in (1.1) by

[g(t)] 1/p [h(t)] x/p [f (t)] v/p and [g(t)] 1/q [h(t)] y/q [f (t)] u/q ,
to obtain the following new inequality: in which x, y, v, u ∈ R and g, f, h are nonnegative real integrable functions such that the involved integrals in (1.2) exist. P.K. Bhandari and S.K. Bissu in [3] applied the inequality (1.2) to establish inequalities for some well-known special functions.
In this paper, we intend to give refinements for inequality (1.1) and (1.2). This is done in Section 2. In section 3, we apply the result obtained in Section 2 to provide refinements to certain inequalities recently obtained by P.K. Bhandari and S.K. Bissu in [3] for the incomplete gamma function, Polygamma functions, Exponential integral function, Abramowitz function, Hurwitz-Lerch zeta function and for the normalizing constant of the generalized inverse Gaussian distribution and the Remainder of the Binet's first formula for ln Γ(x).

Some refinements to Hölder's inequality
M. Akkouchi and M.A. Ighachane in [2] proved the following refinements to Hölder's inequality: Theorem 2.1 ( [2]). Let f 1 and f 2 be real or complex measurable functions on Ω such that ||f 1 || p = 0 and ||f 2 || q = 0. Then for all integers n ≥ 2 we have: As an application of Theorem 2.1, we obtain the following refinements to the inequality (1.1).
Theorem 2.2. Let x, y, v, u ∈ R and f, h be real and nonnegative integrable functions on Ω. For almost all t ∈ Ω, set f Proof. We apply Theorem 2.1 to the measurable functions As a consequence, we have the following corollary.

The Results
In this section, we apply Theorem 2.2 to refine some inequalities established by P.K. Bhandari and S.K. Bissu in [3] for several well-known special functions.

Refinements of an inequality for the Polygamma function
We use Γ(x) to designate the usual gamma function. The Psi function is defined for all x > 0, by Ψ(x) := d dx ln(Γ(x)). For every positive integer m, the Polygamma function Ψ (m) (see [9]) has the following integral representation: For the sequel, for every positive number β, we use the following notation: In [3], the following result was established.
Theorem 3.1 ( [3]). Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If Ψ (m) denotes the Polygamma function, then for all real numbers x, y ∈ (0, ∞) and for all integers u, v ≥ 1 such that v/p + u/q is an integer, we have: We point out that the same arguments of proof in [3] show that the following result is true.
Corollary 3.1. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. Then for all real numbers x, y ∈ (0, ∞) and for all positive numbers u, v, we have: Unauthentifiziert | Heruntergeladen 20.03.20 14:15 UTC By using Theorem 2.2, we obtain the following refinements of the inequality (3.1): Theorem 3.2. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If Ψ (m) denotes the Polygamma function, then for all real number x, y ∈ (0, ∞), for all integers u, v ≥ 1 such that v/p + u/q is an integer and all integers n ≥ 2, we have: Proof. We apply Theorem 2.2, by taking Ω := (0, +∞) equipped with the measure dµ(t) := g(t)dt, where g(t) = 1 1−e −t , and considering the measurable functions f (t) = t and h(t) = e −t . We have the following equalities: Therefore, by Theorem 2.2 we obtain This ends the proof.
As a consequence, we have the following result concerning the associated functions Ψ β (β > 0). Theorem 3.3. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. Then for all real numbers x, y ∈ (0, ∞) and for all positive numbers u, v, we have:

A refined inequality for the incomplete gamma function
We recall (see for example [6]) that the incomplete gamma function is defined for u, x > 0 as It is easy to observe that γ(u, x) is given by the following integral:  ). Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If γ denotes the incomplete gamma function, then for all real numbers x, y ∈ (0, ∞) and for all u, v ≥ 0, we have: By using Theorem 2.2, we obtain the following refinements of the inequality (3.2): Theorem 3.5. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If γ denotes the incomplete gamma function, then for all real numbers x, y ∈ (0, ∞), for all u, v ≥ 0 and all integers n ≥ 2, we have: Proof. We apply Theorem 2.2, by setting Ω := (0, 1) and considering the measure dµ(t) := g(t)dt with g(t) = 1 t for all t ∈ (0, 1) and choosing the functions f (t) = t and h(t) = e −t . We have the following equalities: Therefore, by virtue of Theorem 2.2 and after some easy computations, we obtain which is the desired inequality.

Refinements to certain inequalities for the exponential integral functions
We recall (see [1]) that the Exponential function E n is given by the following integral representation: For the sequel, we need to extend the definition above for all nonnegative real numbers. So, we consider the functions E β (β ∈ R + ) defined by We call them Exponential integral type functions or more simply Exponential integral functions.
Theorem 3.6 ( [3]). Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If E n denotes the Exponential integral function, then for all real number x, y ∈ (0, ∞) and for all integers u, v ≥ 0 such that v/p + u/q is integer, we have: By using Theorem 2.2, we obtain the following refinements of the inequality (3.3): Theorem 3.7. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If E β denotes the Exponential integral function, then for all real number x, y ∈ (0, ∞), for all nonnegative real numbers u, v and all integers n ≥ 2, we have: Proof. We apply Theorem 2.2, by setting Ω := [1, +∞) and considering the measure dµ(t) := g(t)dt, where g(t) = 1, for all t ∈ [1, +∞) and taking f (t) = t −1 and h(t) = e −t for all t ∈ [1, +∞). We have: Then by Theorem 2.2, we have which is the desired inequality.
The following inequality was established in [3].
By some easy computations, we have the following successive equalities: Then by virtue of Theorem 2.2, we have which is the desired inequality.

Refinements of an inequality for the Abramowitz function
The Abramowitz function f n (see [1]) is given for every nonnegative integer and all nonnegative real number x ≥ 0, by We point out that the Abramowitz function has been used in many fields of physics, as the field of particle and radiation transform (see [4] for more details).
For the sequel, we need to extend the definition above to all nonnegative real numbers. So, we consider the functions f β (β ∈ R + ) defined by We call them generalized Abramowitz functions, or more simply, Abramowitz functions.
The following inequality was established in [3].
Theorem 3.10 ( [3]). Let p, q > 1 be real numbers satisfying 1/p+1/q = 1. If f n denotes the Abramowitz function, then for all real numbers x, y ≥ 0 and for all nonnegative integers u, v such that v/p + u/q is an integer, we have: By using Theorem 2.2, we obtain the following refinements of the inequality (3.5): Theorem 3.11. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If f β denotes the Abramowitz function, then for all real numbers x, y ≥ 0, for all nonnegative integers u, v such that v/p + u/q is an integer and all integers n ≥ 2, we have: Proof. We apply Theorem 2.2, by setting Ω := (0, +∞) and considering the measure dµ(t) := g(t)dt, where g(t) = e −t 2 for all t in (0, +∞) and choosing the functions: f (t) = t and h(t) = e −t −1 for all t ∈ (0, +∞). After some easy computations, we have the following equalities: Then by application of Theorem 2.2, we obtain which is the required inequality.

A refined inequality for the normalizing constant of the generalized inverse Gaussian distribution
The generalized inverse Gaussian distribution function (see [7]) is defined for all t > 0 as The number I(α; β, γ) is the normalizing constant, that is: The following inequality was established in [3].

3.7.
A refined inequality for the n-th derivative of the remainder of the Binet's first Formula for ln Γ(x) The Binet's first formula for ln Γ(x) is given by for all number x > 0, where the function is known as the the remainder of the Binet's first Formula for ln Γ(x) (see for example the handbook [1]). By making derivatives, we obtain for every positive integer m ≥ 1: where the function is defined for all number β ≥ 1 and for all positive number x > 0.
The following result was established in [3].
Theorem 3.14 ( [3]). Let p, q > 1 be real numbers satisfying 1/p+1/q = 1. If θ (m) denotes the m-th derivative of the remainder of the Binet's first formula for the logarithm of the function, i.e. ln Γ(x), then for all real numbers x, y ∈ (0, ∞) and for all integers u, v ≥ 1 such that m := v/p + u/q is an integer, we have: Before giving our result, we need the following lemma.
By using Theorem 2.2 and the lemma above, we obtain the following refinements of the inequality (3.8): Theorem 3.15. Let p, q > 1 be real numbers satisfying 1/p + 1/q = 1. If θ (m) denotes the m−th derivative of the remainder of the Binet's first formula for the logarithm of the function, i.e. ln Γ(x), then for all real numbers x, y ∈ (0, ∞), for all integers u, v ≥ 1 such that m := v/p + u/q is an integer and all integers n ≥ 2, we have: Proof. We apply Theorem 2.2, by setting Ω := (0, +∞) and considering the measure dµ(t) := g(t)dt, where g(t) = 1 t 1 e t −1 − 1 t + 1 2 , for all t in (0, +∞) and choosing f (t) = t and h(t) = e −t , for all t ∈ (0, +∞). Note that which is the desired inequality (3.9).