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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