Keywords: increasing monotonicity, sequence, gamma function, ratio of two gamma functions, inequality, logarithmically completely monotonic function, probability of intersecting between a plane couple and a convex body
Turkish Journal of Analysis and Number Theory, 2015 3 (1),
pp 2123.
DOI: 10.12691/tjant315
Received December 25, 2014; Revised February 01, 2015; Accepted February 09, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
On 19 December 2014, Mr. YanZong Zhang, a mathematician in China, asked online a question: is the sequence
 (1.1) 
increasing? if not, can one take an example? where denotes the set of all positive integers. On 20 December 2014, he told that this problem is needed by his teacher, Ms. Jun Jiang, and she said that this problem originates from computation of the probability of intersecting between a plane couple and a convex body in an unpublished paper.
The main of this paper is to give an affirmative answer to the above question.
Theorem 1.1. The sequence defined by (1.1) is strictly increasing.
2. A Direct Proof of Theorem 1.1
We firstly affirm the above question directly.
It is well known that the formula
is called the Wallis sine (cosine) formula, see [^{[9]}, Section 1.1.3], where
is the classical Euler gamma function.
It is clear that and for .
One may reformulate the sequence for in terms of the Euler gamma function as
for . Hence, in order to make sure the increasing monotonicity of the sequences and , it is sufficient to make clear the monotonicity property of the sequence
 (2.1) 
Taking the logarithm of gives
and using the functional equation leads to
As a result, it suffices to prove which is equivalent to
that is,
 (2.2) 
In [^{[7]}, p. 645], Gurland obtained that
 (2.3) 
Later, Chu recovered the ineuqlaity (2.3) in [^{[4]}, Theorem 2]. Since
is equivalent to
the inequality (2.2) is valid. This implies that the sequence , and then the sequence , is strictly increasing. The proof of Theorem 1.1 is complete.
3. The First Indirect Proof of Theorem 1.1
Now we are in a position to give the first indirect proof of Theorem 1.1.
One may observe that the sequence defined by (2.1) may be rearranged as
and
 (3.1) 
where
 (3.2) 
Recall from ^{[1, 11]} that a positive function is said to be logarithmically completely monotonic on an interval if has derivatives of all orders on and its logarithm satisfies for all on . For more information about the notion “logarithmically completely monotonic function”, please refer to ^{[2, 5, 13, 16, 17, 18]} and closely related references therein.
In 1986, J. Bustoz and M. E. H. Ismail revealed in essence in ^{[3]} that
(1) the function
for is logarithmically completely monotonic on if , so is the reciprocal of the function on if ;
(2) the function
for is logarithmically completely monotonic on the interval if , so is the reciprocal of the function on if .
The logarithmically complete monotonicity of the function and imply the strictly increasing monotonicity of the function on . Therefore, by the relation (3.1), the sequence , and then the sequence , is strictly increasing. The proof of Theorem 1.1 is complete.
4. The Second Indirect Proof of Theorem 1.1
Finally we give the second indirect proof of Theorem 1.1.
For real numbers , , and , denote and let
In [^{[12]}, Theorem 1], Qi and Guo discovered the following necessary and sufficient conditions:
(1) the function is logarithmically completely monotonic on if and only if
(2) the function is logarithmically completely monotonic on if and only if
This means that the function
is strictly increasing on , where is defined by (3.2). As a result, by the relation (3.1), the sequence , and so the sequence , is strictly increasing. The proof of Theorem 1.1 is complete.
Remark 4.1. The reciprocal of the sequence for is a (logarithmically) completely monotonic sequence. For information on the definition of (logarithmi cally) completely monotonic sequences and related properties, please refer to closely related chapters in the books ^{[8, 19]}.
Remark 4.2. By carefully reading the expository and survey articles ^{[9, 10, 14, 15]} and a large amount of references therein, one may deeply understand and exten sively comprehend the spirit and essence of this paper.
Remark 4.3. This paper is a slightly revised version of the preprint ^{[6]}.
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