## AN ALTERNATIVE PROOF OF ELEZOVIĆ-GIORDANO-PEČARIĆ’S THEOREM (903)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Qi903analternative,

author = {Feng Qi and Bai-ni Guo},

title = {AN ALTERNATIVE PROOF OF ELEZOVIĆ-GIORDANO-PEČARIĆ’S THEOREM},

year = {903}

}

### OpenURL

### Abstract

Abstract. In the present note, an alternative proof is supplied for Theorem 1

### Citations

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27 |
Some extensions of W. Gautschis inequalities for the gamma function
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(Show Context)
Citation Context .... The function zs,t(x) is either convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1. The explicit or implicit origins and background of this theorem may be traced back to =-=[3, 5, 18, 20]-=- and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, [1, 6, 8, 11, 16, 17, 18]. For detailed information on its history, pleas... |

17 |
Some elementary inequalities relating to the gamma and incomplete gamma functions
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Citation Context .... The function zs,t(x) is either convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1. The explicit or implicit origins and background of this theorem may be traced back to =-=[3, 5, 18, 20]-=- and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, [1, 6, 8, 11, 16, 17, 18]. For detailed information on its history, pleas... |

17 | Note on a class of completely monotonic functions involving the polygamma functions, Tamsui Oxf
- Qi, Guo, et al.
(Show Context)
Citation Context ...igins and background of this theorem may be traced back to [3, 5, 18, 20] and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, =-=[1, 6, 8, 11, 16, 17, 18]-=-. For detailed information on its history, please refer to the survey article [9] published as a preprint recently. The purpose of this note is to supply an alternative proof for Theorem 1. 2. Lemmas ... |

17 | A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality - QI, GUO - 2007 |

16 | The best bounds in Gautschi’s inequality - ELEZOVIĆ, GIORDANO, et al. |

12 |
The best bounds in Kershaw’s inequality and two completely monotonic functions
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Citation Context ...igins and background of this theorem may be traced back to [3, 5, 18, 20] and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, =-=[1, 6, 8, 11, 16, 17, 18]-=-. For detailed information on its history, please refer to the survey article [9] published as a preprint recently. The purpose of this note is to supply an alternative proof for Theorem 1. 2. Lemmas ... |

11 |
Bounds for the Ratio of Two Gamma Functions
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Citation Context ...rem or its special cases have been proved several times by different approaches in, for example, [1, 6, 8, 11, 16, 17, 18]. For detailed information on its history, please refer to the survey article =-=[9]-=- published as a preprint recently. The purpose of this note is to supply an alternative proof for Theorem 1. 2. Lemmas In order to prove Theorem 1 alternatively, the following lemmas are necessary. 20... |

10 |
The best bounds in Gautschi-Kershaw inequalities
- QI, GUO, et al.
(Show Context)
Citation Context ...igins and background of this theorem may be traced back to [3, 5, 18, 20] and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, =-=[1, 6, 8, 11, 16, 17, 18]-=-. For detailed information on its history, please refer to the survey article [9] published as a preprint recently. The purpose of this note is to supply an alternative proof for Theorem 1. 2. Lemmas ... |

7 |
A note on gamma function
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- 1959
(Show Context)
Citation Context .... The function zs,t(x) is either convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1. The explicit or implicit origins and background of this theorem may be traced back to =-=[3, 5, 18, 20]-=- and [6, Theorem 2]. This theorem or its special cases have been proved several times by different approaches in, for example, [1, 6, 8, 11, 16, 17, 18]. For detailed information on its history, pleas... |

7 |
Note on the gamma function
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Citation Context |

5 | Monotonicity and convexity for the gamma function
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5 |
Functional equations for Wallis and Gamma
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5 | Three-log-convexity for a class of elementary functions involving exponential function - QI |

4 |
Properties and applications of a function involving exponential functions
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(Show Context)
Citation Context ...rem 1, since properties of qα,β(u) on (−∞, 0) are idle there. Remark 2. The logarithmically convex properties in Lemma 3 of this paper were also proved in [10] by using different techniques. Also see =-=[4]-=- and related references therein. Remark 3. It is well-known that a positive and k-times differentiable function f(x) is said to be k-log-convex (or k-log-concave, respectively) on an interval I with k... |

4 |
Laplace Transform, From MathWorld—A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/LaplaceTransform.html
- Weisstein
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Citation Context ...et using AMS-LATEX. 12 F. QI AND B.-N. GUO Lemma 1 ([7, p. 16]). The polygamma functions ψ (n)(x) can be expressed for x > 0 and n ∈ N as ψ (n) (x) = (−1) n+1 ∫ ∞ 0 t n 1 − e −te−xt dt. (3) Lemma 2 (=-=[19]-=-). Let fi(t) for i = 1, 2 be piecewise continuous in arbitrary finite intervals included on (0, ∞), suppose there exist some constants Mi > 0 and ci ≥ 0 such that |fi(t)| ≤ Miecit for i = 1, 2. Then ∫... |

3 |
Monotonicity and logarithmic convexity for a class of elementary functions involving the exponential function
- Qi
(Show Context)
Citation Context ...ss of the proof provided in [16, 17] for Theorem 1, since properties of qα,β(u) on (−∞, 0) are idle there. Remark 2. The logarithmically convex properties in Lemma 3 of this paper were also proved in =-=[10]-=- by using different techniques. Also see [4] and related references therein. Remark 3. It is well-known that a positive and k-times differentiable function f(x) is said to be k-log-convex (or k-log-co... |