An inequality for increasing sequences and its integral analogue

✍ Scribed by Horst Alzer

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 204 KB
• Volume: 133
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90034-5

✦ Synopsis

We prove the following results: (i) Let p> 1 be a real number and let n>2 be an integer. If (Ui) (i=O,l,..., n) is an increasing sequence of real numbers, then i: iai-oi-l)a'l+l-i~c~,~[ i~l(ui-ai-l~a~~l-i ' 1 (aO=o) i=l holds with Cn,p= 1. (ii) Let p > 1 be a real number. IffE L' [0, l] is nonnegative, then j+) ( I;-'/(Od~)'dxBC,{ ~~~(x)(~~-*~(r)dl>"'dx~' (*) holds with C,= 1. The constants C,_,= 1 and C,= 1 are the best possible. Inequality (*) extends a result of SzCkely et al. (1992).