包含q-psi函數的函數完全單調性及其應用

汪韓,王連堂

(西北大學數學學院,陜西 西安 710127)

包含q-psi函數的函數完全單調性及其應用

汪韓,王連堂

(西北大學數學學院,陜西 西安 710127)

主要證明了涉及q-digamma函數的完全單調性.通過引入經典q-理論將包含digamma函數的函數進行q-模擬,利用q-模擬函數以及級數的性質,得到了包含q-digamma函數的完全單調性.最后利用它們的完全單調性得到了有關q-digamma和q-trigamma函數的不等式.

完全單調;q-模擬;q-psi函數;不等式;q-trigamma函數

1 引言

Gamma函數與psi(digamma)函數在x＞0時的定義為:

psi函數是Gamma函數的對數導數,psi函數的各階導數ψ(i)(x)對于i∈N, N={1,2,3,…},被稱為polygamma函數在文獻[1]中,定義了x＞0時.Gamma函數的q-模擬

q-psi函數為q-Gamma函數的對數導數,

psi函數和q-psi函數被包含在很多不等式中,見文獻[2-3],由(1)和(2)可以得到如下：

當0＜q＜1且x＞0,

當q＞1且x＞0,

ψ(k)(x)的q-模擬為k∈N,被稱為q-polygamma函數,其中和分別被稱為q-trigamma函數和q-tetragamma函數對(3)式兩邊取對數后求導可得,當q＞0時,有

文獻[4-5]中介紹了gamma函數及psi函數和它們的q-模擬之間的關系.

更多關于q-gamma函數的內容,見文獻[6-8].

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文獻[9-10]給出了完全單調性的定義:一個函數f被稱為區間I上的完全單調函數,如果f在區間I上的各階導數滿足對于任意的x∈I且n≥0,有(?1)nf(n)(x)≥0.

完全單調函數在各個分支學科都有應用,如,解析數論、概率論、物理學,見文獻[11-13].文獻[14]證明了函數

在(0,∞)上是完全單調的,當且僅當

當x＞0時,定義函數

它們的q-模擬函數為:

容易得到

(9)式的q-模擬函數的完全單調性在文獻[16-18]中被證明.近年來,研究包含psi和q-psi函數完全單調性及不等式的文獻越來越多,Gamma函數、psi函數以及他們的q-模擬的許多性質和不等式在文獻[19–23]中被得到.

本文主要證明了fq(x),Fq(x),gq(x),Gq(x)的完全單調性,并由它們的完全單調性得到了關于q-psi函數和q-polygamma函數的不等式.

2 引理及其證明

引理 1當x＞0時,對i∈N及q∈(0,1),有

證明對(1.4)式直接求導可得到.

引理 2當x＞0,q∈(0,1)時,有

證明由(1.4)式可知,

引理 3當0＜p＜1時,定義函數

則hp(x)及ηp(x)在(0,∞)上大于零且單調遞增.

引理 4 當0＜q＜1,對i∈N,有下面式子成立

其中fq(x)和gq(x)由(11)和(13)定義.

3 主要結論

定理 3.1當0＜q＜1時,(11)定義的函數fq(x)在(0,∞)上是完全單調的.

定理3.2當0＜q＜1時,(13)定義的函數gq(x)的二階導數在(0,∞)上是完全單調的.

推論 3.3當q＞1時,(12)定義的函數 Fq(x)及(14)定義的函數 Gq(x)的二階導數在(0,∞)上是完全單調的.

推論 3.4當x∈(0,∞)時有下面不等式成立

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Complete monotonicity functions involving the q-psi function and some applications

Wang Han,Wang Liantang
(College of Mathematics,Northwest University,Xi′an 710127,China)

In this paper,the complete monotonicity for functions involving q-digamma functions are proved, some applications of these results give inequalities containing q-digamma and q-trigamma functions.

complete monotonicity,q-analogue,q-psi function,inequality,q-trigamma function

O174.6

A

1008-5513(2017)01-0082-10

10.3969/j.issn.1008-5513.2017.01.009

2016-07-13.

陜西省自然科學基金(2010JM1017).

汪韓(1992-),碩士生,研究方向：特殊函數論.

2010 MSC:26A48