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研究生:李嘉修
研究生(外文):Jia-shiou Li
論文名稱:有關具對數型凹函數正數列的不等式
論文名稱(外文):On inequalities of positive and logarithmically convex sequences
指導教授:楊國勝楊國勝引用關係
指導教授(外文):Gou-sheng Yang
學位類別:碩士
校院名稱:淡江大學
系所名稱:數學學系碩士班
學門:教育學門
學類:普通科目教育學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:中文
論文頁數:38
中文關鍵詞:單調不等式幾何平均比率正序列對數型凸函數對數型凹函數數學歸納法
外文關鍵詞:monotonicityinequalitygeometric meamratiopositive sequencelogarithmically concavelogarithmically convexmathemathical induction
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在本文中我們推廣、整理了ㄧ些BAI-NI Gao 及 FENG QI所證明的不等式。
In the present note we established a generalization some of the inequalities that proof by BAI-NI Gao AND FENG QI.
目 錄

1.簡介…………………………………………… P1.

2.主要結果……………………………………… P1.
定理1……………………………………………P1.
定理2……………………………………………P6.
定理3……………………………………………P10.

3.應用…………………………………………… P15.

4.參考文獻……………………………………… P17.

Contents

1.Introduction…………………………P20.

2.Main results.…………………………P20.
Theorem 1…………………………………P20.
Theorem 2…………………………………P25.
Theorem 3…………………………………P29.

3.Applications…………………………P34.

4.References……………………………P36.
[1] BAI-NI Gao AND FENG QI, Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity, Mathematicity Inequalities & Applications, 9, 1 (2006). 1-9.
[2] H. ALZER, On an inequality of H. Minc and L. Sathre, J. Math. Anal.Appl., 179, (1993), 396-402.
[3] B.-N. Guo, F. QI, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math., 7, 2 (2003), 239-247.
[4] T. H. Chan, P. Gao and F. QI, On a generalization of Martins'' inequality,Monatsh. Math., 138, 3 (2003), 179-187. RGMIA Res. Rep. Coll., 4, 1 (2001), Art.12, 93-101. Available online at URL: http://rgmia.vu.edu.au/v4n1.html.
[5] Chao-Ping Chen, Feng QI, P. Cerone and S.S. Dragomir, Monotonicity of Sequences Involving Convex and Concave Functions, Math. Inequal. Appl. 6, 2 (2003), 229-239. RGMIA Res. Rep. Coll., 5, 1 (2002), Art 1,3-13. Available online at URL: http://rgmia.vu.edu.au/v5n1.html.
[6] D. KERSHAW, A. LAFORGIA, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. SCI. Fis. Mat. Natur., 119, (1985),127-133.
[7] J.-CH. KUANG, Some extensions and refinements of Minc-Sathre inequality, Math. Gaz., 83, (1999), 123-127.
[8] H. MINC, L. SATHRE, Some inequalities involving (r!)^{(1/r)}, Proc. Edinburgh Math. Soc., 14, (1964/65), 41-46.
[9] J. PEČARIĆ, F. PROSCHAN, AND Y. L. TONG, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, 187, Academic Press, 1992.
[10] F. QI, An algebraic inequality, J. Inequal. Pure Appl. Math., 2, 1 (2001), Art. 13. Available online at URL: http://jipam .vu.edu.au/artical.php?sid=129. RGMIA Res Rep. Coll., 2, 1 (1999), Art. 8, 81-83. Available
online at URL: http://rgmia.vu.edu.au/v2n1.html.
[11] F. QI, Generalization of H. Alzer''s inequality, J. Math. Anal. Appl.,240, (1999), 294-297.
[12] F. QI, Inequalities and monotonicity of sequences involving [n]√((n+k)!/k!),Soochow J. Math., 29, 4 (2004), 353-361. RGMIA Res. Rep. Coll., 2,5 (1999), Art. 8, 685-692. Available online at URL: http://rgmia.vu.edu.au/v2n5.html.
[13] F. QI, Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with unit difference, Internat. J. Math. Ed. Sci. Tech., 34, 4 (2003), 601-607. Austral. Math Soc. Gaz., 30, 3 (2003), 142-147. RGMIA Res. Rep. Coll., 6, (2003), suppl., Art. 2. Available online at URL: http://rgmia.vu.edu.au/v6(E).html
[14] B.-N. GUO, F. QI, Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference, Tamkang. J. Math., 34, 3 (2003), 261-270.
[15] F. QI, On a new generalization of Martin''s inequality, RGMIA Res. Rep. Coll., 5, 3 (2002), Art. 13, 527-578. Available online at URL: http://rgmia.vu.edu.au/v5n3.html.
[16] F. QI, B.-N. GUO, An inequality between ratio of the extended logarithmic means and ratio of the exponential means, Taiwanese J. Math., 7,2 (2003), 229-237. RGMIA Res. Rep. Coll., 4, 1 (2001), Art. 8, 55-61. Available online at URL: http://rgmia.vu.edu.au/v4n1.html.
[17] F. QI, B.-N. GUO, Monotonicity of sequences involving convex function and sequence, Math. Inequal. Appl. (2006), in press. RGMIA Res. Rep. Coll., 3, 2 (2000), Art. 14, 321-329. Available online at URL: http:// rgmia.vu.edu.au/v5n3.html.
[18] F. QI, B.-N. GUO, Monotonicity of sequences involving geometric means of postive sequences with logarithmical convexity, RGMIA Res. Rep. Coll., 5, 3 (2002), Art. 10, 497-507. Available online at URL: http:// rgmia.vu.edu.au/v3n2.html.
[19] F. QI, B.-N. GUO, Some inequalities involving the geometric mean of natural numbers and the ratio of gamma functions, RGMIA Res. Rep. Coll., 4, 1 (2001), Art. 6, 41-48. Available online at URL: http://rgmia.vu.edu.au/v4n1.html.
[20] F. QI, Q.-M. LUO, Generalization of H. Minc and J. Satnre''s inequality, Tamkang J. Math., 31, 2 (2000), 145-148. RGMIA Res. Rep. Coll., 2, 6 (1999), Art. 14, 909-912. Available online at URL: http://rgmia.vu.edu.au/v2n6.html.
[21] F. QI, N. TOWGHI, Inequalities for the ratios of the mean values of functions, Nonlinear Funct. Anal. Appl., 9, 1 (2004), 15-23. An inequality for the ratios of the arithmetic means of functions with a positive parameter, RGMIA Res. Rep. Coll., 4, 2 (2001), Art. 15, 305- 309. Available online at URL:http://rgmia.vu.edu.au/v4n2.html.
[22] J. A. SAMPAIO MARTINS, Inequalities of Rado-Popoviciu type, In: Marques de Sá, Eduardo (ed.) et al. Mathematical studies. Homage to Professor Doctor Lús de Albuquerque. Coimbra: Universidade de Coimbra, Faculdade de Ciências e Tecnoligia, Departamento de Matemática, 169-175 (1994).
[23] J. SÁNDOR, On the gamma function, I-III, Publ. C. R. M. P. Neuchâtel,Série 1, 21, (1989), 4-7; Série 1, 28, (1997), 10-12; Série 2, 19 (2001), 33-40.
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