HE DI ELMONSER. 1. Introduction In 1964 H. Mink and L. Sathre [15] proved the following inequality. n, n N. ((n + 1)!) n+1

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1 -ANALOGUE OF THE ALZER S INEQUALITY HE DI ELMONSER Abstact In this aticle, we ae inteested in giving a -analogue of the Alze s ineuality Mathematics Subject Classification (200): 26D5 Keywods: Alze s ineuality; -analogue Aticle histoy: Received 26 Apil 205 Received in evised fom 24 August 205 Accepted 26 August 205 Intoduction In 964 H Mink and L Sathe [5] poved the following ineuality () (n!) n n, n N < n+ ((n + )!) n+ The ineuality () was genealized and efined by H Alze in [2]-[4] He poved in [4] the following ineuality: (2) # " n (n + ) i i (n!) n n, n N, R+ < n+ n+ n i i ((n + )!) n+ The lowe and uppe bounds ae the best possible Many poofs of the ineuality (2) and some genealizations wee given in ([],[5]-[7],[9],[0],[2] -[4],[6]-[23]) The left hand side of the Alze s ineuality (2) was genealized by Feng Qi [8] as follows: (3) n+m+k " n n+m ik+ i n+m+k ik+ i #, n, m N, whee k is a nonnegative intege and R+ The lowe bound is best possible The main pupose of this pape is to give a -analogue of ineualities (2) and (3) 200 Mathematics Subject Classification 26D5 Key wods and phases Alze s ineuality; -analogue 45

2 2 -analogue of the Alze s ineuality Thoughout this pape, we conside a positive intege 6 and fo x C, we wite (2) [x] x Note that [x] tends to x when tends to (we efe to [] fo moe details about -calculus) To pove the main esult of the pape, we need the following lemma Lemma 2 Fo all, fo all nonnegative integes n and k and fo all nonnegative eal numbe, we have (22) [i] ik+ [n] [n + k] [n + k + ] [n + ] [n + k + ] [n] [n + k] oof Let k be a nonnegative intege and be a nonnegative eal numbe We pove the esult by induction on n Fo n, we have [k + ] [k + ] [k + 2] [2] [k + 2] [k + ] [k + ] [2] [k + 2] [k + ] [k + 2] [2] [k + 2] [k + ] [k + ] [k + 2] [k + ] ( + )[k + 2] [k + ] [k + ] ( k+2 ) ( k+ ) ( + )( k+2 ) ( k+ ) ] k+ Using the fact that k+ k+2 [k + ) ( k+2 k+ ) + ( k+2 < <, we get [k + ] [k + ] [k + 2], [2] [k + 2] [k + ] which achieves the poof of the esult fo n Suppose, now, that it is valid fo n and let s pove that it s valid fo n + Using the fact + that ik+ [i] ik+ [i] + [n + k + ], calculating staightfowadly, and simplifying easily, the induction step can be witten as [n + 2] [n + ] [n + k + ] [n + ] [n + k + ] [n] [n + k] Conside the functions f and g defined on [n, n + ] as follows f (x) [x + ] [x + k + ] and [n + k + ] g(x) [x] [x + k] Simple deivation gives f (x) ln x+ [ [x + k + ] + x+k+ [x + ] [x + k + ] ] and g (x) ln x [ [x + k] + x+k [x] [x + k] ] 46

3 So, fo all x [n, n + ], we have f (x) g (x) x+ [x + k + ] + x+k+ [x + ] [x + k + ] x [x + k] + x+k [x] [x + k] [x+] [x + k + ] + k [x+k+] [x] k [x + k] + [x+k] [x + k + ] [x + k] Fom Cauchy s mean-value theoem and the pevious ineuality, thee exists one point ξ (n, n + ) such that [n + 2] [n + ] [n + k + ] f (ξ) [ξ + k + ] (23) [n + ] [n + k + ] [n] [n + k] g (ξ) [ξ + k] But, [ξ + k + ] [ξ + k] [n + k + ] ( ξ+k+ )( + ) ( +2 )( ξ+k ) ( ξ+k )( + ) +2 + ξ+k ξ+k+ + ( ξ+k )( + ) k ( )( n+ ξ ) 0 ( ξ+k )( + ) Then, [ξ + k + ] [ξ + k] This ineuality togethe with (23) gives (24) [n + k + ] [n + 2] [n + ] [n + k + ] [n + ] [n + k + ] [n] [n + k] [n + k + ], which poves that the esult is valid fo n + Now, we ae in a situation to pove the main esult of this pape Theoem 22 (25) i [i] [n+m] ik+ +m [n] ik+ i [i], if ]0, [, [n + k] m(+ ) [n + m + k] [n+m] ik+ [i] +m, if ], + [, [i] [n] ik+ whee n, m N, k is a nonnegative intege and R+ The lowe bounds ae best possible oof It is easy to veify that fo all positive eal 6, we have [n] n [n] and so, [n + k] [n + k] [n + k] m n+m+k [n + m + k] [n + m + k] [n + m + k] Then, to pove the esult, it suffices to focus on the case 47

4 Let and be a nonnegative eal numbe Fom the pevious lemma and the fact that + [i] (26) [i] + [n + k + ] ik+ ik+ we obtain fo all n N and k nonnegative intege + [i] [i] [n] [n + k] [n + ] [n + k + ] (27) ik+ ik+ So, by induction on m, we get fo all n N and k, m nonnegative integes [i] [n] [n + k] [n + m] [n + m + k] ik+ n+m+k [i] ik+ Then, [n + k] [n + m + k] which achieves the poof [n + m] ik+ [i] <, +m [n] ik+ [i] The it case is given by ], + [, (28)! [n + m] ik+ [i] [n + k] +m [n + m + k] [n] ik+ [i] + ]0, [, (29) + m(+ )! [n + m] ik+ i [i] [n + k] +m i [n + m + k] [n] ik+ [i] Thus, the lowe bound is best possible [i] Indeed, using the fact that 0 [j] <, i < j, ], + [, +! [n + m] ik+ [i] +m [n] ik+ [i] + [n + m] [n] [n + k] [n + m + k] 48 [i] [n + k] + ik+ ( [] ) [n + m + k] + +m ( [i] ) ik+ [n+m+k]

5 ]0, [, + m(+ )! [n + m] ik+ i [i] +m [n] ik+ i [i] + m(+ ) + + [n + m] [n] [n + k] [n + m + k] [i] ik+ ( [] ) +m ik+ [i] ( [n+m+k] ) [n + k] [n + k] m [n + m + k] [n + m + k] Fo k 0 and m, we find the following special case: Coollay 23 If, ae positive eal numbes and n is a positive intege, then i [i] [n+] n i, if ]0, [, i [i] [n] n+ [n] + i (20) [n + ] [n+] n i [i] n+, if ], + [ [n] i [i] The lowe bounds ae best possible Remak 24 When tends to ( + o ), [n] tends to n and the ineuality (20) tends to the Alze s one Refeences [] S Abamovich, J Baic, M Matic and J ec aic, On van de Lune-Alzes ineuality, J Math Ineual (2007), no 4, [2] Host Alze, On some ineulities involving (n!) n, Rocky Mt J Math 24 (994), no 3, [3] Host Alze, On some ineulities involving (n!) n,ii, eiod Math Hung 28 (994), no 3, [4] Host Alze, On an ineuality of H Minc and L Sathe, J Math Anal Appl 79 (993), [5] G Bennett, Meaningful seuences, Houston J Math, 33(2007), no2, [6] S S Dagomi and J Van Hoek, Some new analytic ineualities and thei applications in guessing theoy, J Math Anal Appl 225 (998), Issue 2, [7] N Elezovic and J ecaic, On Alze z ineuality, J Math Anal Appl 223(998), [8] Feng Qi, Genealization of H Alze s Ineuality, J Math Anal Appl 240 (999), [9] Feng Qi, Genealization of H Alze s and Kuang s ineuality, Tamkang J Math 3 (2000), no 3 [0] Feng Qi and L Debnath, On a new genealization of Alze s ineuality, Intenat J Math and Math Sc 23 (2000), no 2, (999), no 6, Aticle 4 [] G Gaspe and M Rahman, Basic Hypegeometic Seies, Encyclopedia of Mathematics and its application, Vol 35, Cambidge Univ ess, Cambidge, UK, 990 [2] J C Kuang, Some extensions and efinements of Minc-Sathe ineuality, Math Gaz 83 (999), [3] Z Liu, New genealizations of H Alze s ineuality, Tamkang J Math 34(2003), no3, [4] J S Matins, Aithmetic and geometic means, an application to Loentz seuence spaces, Math Nach 39 (988), [5] H Mink and L Sathe, Some ineualities involving (!), oc Edinbugh Math Soc 4 (964965), 4-46 [6] N Ozeki, On some ineualities, J College Ats Sci Chiba Univ4 (965), no 3, 2-24 (Japanese) 49

6 [7] Jo zsef Sa ndo, On an ineuality of Alze, J Math Anal Appl 92 (995), [8] Jo zsef Sa ndo, Comments of an ineuality fo the sum of powes of positive numbes, RGMIA Res Rep Coll 2 [9] Jo zsef Sa ndo, On an ineuality of Alze, II, Octogon Math Mag, (2003), No2, (999), no 2, [20] Jo zsef Sa ndo, On an ineuality of Alze fo negative powes, RGMIA, 9(2006), NO4, At4 [2] J S Ume, An elementay poof of Alze s ineuality, Math Japon 44 (996), no 3, [22] J S Ume, An ineuality fo positive eal numbes, MIA, 5(2002), no4, [23] J S Ume, A simple poof of genealized Alze ineuality, Indian J ue Appl Math, 35 (2004), no 8, Institut National des sciences appliues et de technology, Tunis, Tunisia addess: monseu2004@yahoof 50

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics