Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.016927

ARTICLE

Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions

1Faculty of Science, Mathematics Department, Mansoura University, Mansoura, 35516, Egypt

2Faculty of Science, Mathematics Department, Jeddah University, Jeddah, 21589, Saudi Arabia

3Faculty of Science, Mathematics Department, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

*Corresponding Author: Mansour Mahmoud. Email: mansour@mans.edu.eg

Received: 11 April 2021; Accepted: 20 October 2021

Abstract: In this paper, we study the completely monotonic property of two functions involving the function

Keywords: Trigamma function; tetragamma function; completely monotonic function; completely monotonic degree; inequality

The Euler’s gamma function is defined [1] by the improper integral

and the psi or digamma function is defined by the logarithmic derivative of gamma function, that is

The two derivatives

Kirchhoff was the first to apply the polygamma functions

A function f(x) is said to be completely monotonic [6] (Chapter XIII) on an interval J if the derivatives f(m)(x) exist on J for all

According to the Bernstein–Widder theorem [7] the function f(x) is completely monotonic on

where the integral converges for

In 2004, Alzer [8] presented the inequality

where

and

In 2013, Guo et al. [9] proved the complete monotonicity on

In [10], Zhao et al. proved the complete monotonicity on

and

They also presented the double inequality

The lower bound of inequality (2) and the bound of inequality (1) are not included each other.

In 2015, Qi [11] proved the complete monotonicity on

If and only if

If and only if

The survey [12] presented several results about the function

Let the function f (x) be completely monotonic for

Qi [27] proved that

and Guo et al. [26] proved that

where C(x) is a polynomial of positive coefficients of degree twenty one. For more results related to properties of completely monotonic degree, we refer to [12,16,28–32] and the references therein.

Our first aim will be to establish the double inequality

which improves the upper bound of inequality (3) for x > 0 and the lower bound of inequality (3) for

and

The second aim of this paper is to compute the completely monotonic degree of a function involving

For proving our main results, we need the following lemmas.

Lemma 2.1 The function

is completely monotonic on

where

Proof. Using the integral representations formulas [33]

and

we have

where

with

Using the inequalities 3n > n2 for

Also, by using the inequalities

and hence an > 0 for

where the mth Bernoulli number Bm is defined by [34]

We obtain

Lemma 2.2 The function

is completely monotonic on

Proof. Using the formulas (6) and (7), we obtain

where

with

Now

By using that

For

Hence bn < 0 for

Now we begin to prove our main results.

Theorem 3.1 The functions

and

are completely monotonic on

Proof. Using recursion formula [33]

we get

The two functions

We get

Now, using the recursion formula (16), we get

The two functions

The proof of Theorem 1 is complete.

Remark 1. The upper bound of inequality (15) is better than the upper bound of inequality (3) for x > 0. Also, the lower bound of inequality (15) is better than the lower bound of inequality (3) for

Theorem 3.2. The completely monotonic degree of L(x) on

Proof. Using the integral representation (8), we get

where

with

Now using the inequalities

we obtain that kn > 0 for

If we suppose that

where

with

Using the relation [35]

We get

Also,

Then

and hence we get

Combining (18) and (19) completes the proof.

The main conclusions of this paper are stated in Theorems 3.1 and 3.2. Concretely speaking, the authors proved the completely monotonic property of two functions involving the sum of the Trigamma square and Tetragamma functions, derived a new double inequality for this sum and deduced the completely monotonic degree of a function involving the Trigamma function.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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