AN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.4 (7) (2008), 9 96 AN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY CHU YUMING, ZHANG XIAOMING AND TANG XIAOMIN Abstrct. We get n upper bound estimte for H. Alzer s integrl inequlity. As pplictions, we obtin some inequlities for the logrithmic men.. Introduction For b > > 0, the logrithmic men L(, b) of nd b is defined s L(, b) = log b log. (.) The logrithmic men hs numerous pplictions in physics. Mny properties nd inequlities re obtined by mny mthemticins (see [-8] nd the references therein). In 989, H. Alzer [9] proved the following result. Theorem A. Suppose b > > 0, nd f C[, b] is strictly incresing function. If is strictly convex, then f f(x) dx > ()f(l(, b)). (.2) The min purpose of this pper is to get the upper bound estimte for f(x) dx. Our min result is the following Theorem Mthemtics Subject Clssifiction. 26D07. Key words nd phrses. Convex function, integrl inequlity, logrithmic men. This reserch is supported by the NSF of P. R. Chin under Grnt No nd , Foundtion of the Eductionl Committee of Zhejing Province under Grnt No nd by the NSF of Zhejing Rdio nd TV University under Grnt No. XKT07G9.

2 92 CHU YUMING, ZHANG XIAOMING AND TANG XIAOMIN Theorem. Suppose b > > 0, nd f C[, b] is strictly incresing function. If is strictly convex, then f f(x) dx < b(l(, b) )f(b) + (b L(, b))f(). (.3) L(, b) As pplictions for Theorem A nd Theorem, we shll give some inequlities for the logrithmic men in Section Proof of Theorem Proof of Theorem. Put c = f() nd d = f(b). Then f : [c, d] [, b] is strictly incresing. For ny x [0, ], the strict convexity of implies f f [xf() + ( x)f(b)] = f [xc + ( x)d] < x f (c) + x f (d) = x + x. (2.) b Since f is strictly incresing, (2.) leds to ( xf() + ( x)f(b) > f b xb + ( x) ). (2.2) Next, for ny t [, b], tking x = (b t) b t(b ), then 0 x nd t = xb+( x). The inequlity (2.2) nd the trnsformtion of the vrible of integrtion yield f(t) dt = b() < b() = 0 ( f b xb+( x) ) [xb + ( x)] 2 dx f()x + f(b)( x) 0 [()x + ] 2 dx b(l(, b) )f(b) + (b L(, b))f(). L(, b) 3. Applictions In this section, we shll prove number of inequlities for logrithmic men underlying H. Alzer s inequlity. Theorem 2. If b > > 0 nd α > 0, then L(, b) > α + α b(b α α ) b α+ α+ (3.)

3 nd AN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY 93 [ b α+ α+ ] α L(, b) <. (3.2) (α + )() Proof. Tking f(x) = x α, then f : [, b] [ α, b α ] is strictly incresing, nd g = : [ α, b α ] [ f b, ] stisfies is strictly convex on [ α, b α ]. Thus The- for x [ α, b α ]. Eqution (3.3) implies tht orem A nd Theorem imply ()L α (, b) < g (x) = α ( α + )x α 2 > 0 (3.3) f x α dx < bα+ (L(, b) ) + α+ (b L(, b)). (3.4) L(, b) Equtions (3.) nd (3.2) follow from eqution (3.4). The following result is well-known. Theorem 3. If b > > 0, then b < L(, b) < e ( b b ) b. (3.5) Proof. Tking f(x) = log x, then f : [, b] [log, log b] is strictly incresing, nd g = : [log, log b] [ f b, ] stisfies for x [log, log b]. Eqution (3.6), Theorem A nd Theorem yield g (x) = e x > 0 (3.6) b(l(, b) ) log b + (b L(, b)) log () log L(, b) < log x dx <. L(, b) (3.7) Eqution (3.5) follows from eqution (3.7). Theorem 4. If b > > 0, then log eb e < L(, b) < b(e b e ) (b )e b ( )e. (3.8) Proof. Tking f(x) = e x, then f : [, b] [e, e b ] is strictly incresing, nd g = : [e, e b ] [ f b, ] stisfies g (x) = log x + 2 x 2 (log x) 3 > 0 (3.9)

4 94 CHU YUMING, ZHANG XIAOMING AND TANG XIAOMIN for x [e, e b ]. Eqution (3.9), Theorem A nd Theorem yield ()e L(,b) < Eqution (3.8) follows from eqution (3.0). Theorem 5. If π 2 nd L(, b) > e x dx < b(l(, b) )eb + (b L(, b))e. (3.0) L(, b) > b > > 0, then tn L(, b) < log cos log cos b (3.) b(tn b tn ) (b tn b + log cos b) ( tn + log cos ). (3.2) Proof. Tking f(x) = tn x, then f : [, b] [tn, tn b] (0, + ) is strictly incresing, nd g = : [tn, tn b] [ f b, ] stisfies g (x) = 2( + x rctn x) ( + x 2 ) 2 (rctn x) 3 > 0 (3.3) for x [tn, tn b]. Eqution (3.3), Theorem A nd Theorem yield b(l(, b) ) tn b + (b L(, b)) tn (b ) tn L(, b) < tn x dx <. L(, b) (3.4) Equtions (3.) nd (3.2) follow from eqution (3.4). Theorem 6. If b > > 0, then nd rctn L(, b) < b rctn b + 2 log( + 2 ) rctn 2 log( + b2 ) (3.5) L(, b) > 2b(rctn rctn ) log( + b 2 ) log( + 2 ). (3.6) Proof. Tking f(x) = rctn x, then f : [, b] [rctn, rctn b] is strictly incresing, nd g = : [rctn, rctn b] [ f b, ] stisfies for x [rctn, rctn b]. g (x) = 2 csc x cot x > 0 (3.7)

5 AN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY 95 Eqution (3.7), Theorem A nd Theorem yield () rctn L(, b) < < rctn x dx b(l(, b) ) rctn b + (b L(, b)) rctn. (3.8) L(, b) Equtions (3.5) nd (3.6) follow from equtions (3.7) nd (3.8). Theorem 7. If π 3 nd b > > 0, then L(, b) > sin L(, b) < cos cos b (3.9) b(sin b sin ) (b sin b + cos b) ( sin + cos ). (3.20) Proof. Tking f(x) = sin x, then f : [, b] (0, π 3 ] [sin, sin b] (0, 3 is strictly incresing, nd g = : [sin, sin b] (0, 3 f 2 ] [ b, ] stisfies g (x) = 2 x 2 x rcsin x > 0 (3.2) ( x 2 ) 3 2 (rcsin x) ]. for x [sin, sin b] (0, Eqution (3.2), Theorem A nd Theorem yield b(l(, b) ) sin b + (b L(, b)) sin () sin L(, b) < sin x dx <. L(, b) (3.22) Equtions (3.9) nd (3.20) follow from eqution (3.22). Theorem 8. If b > > 0, then nd rcsin L(, b) < b rcsin b + b 2 rcsin 2 L(, b) > 2 ] (3.23) b(rcsin rcsin ) 2 b 2. (3.24) Proof. Tking f(x) = rcsin x, then f : [, b] (0, ] [rcsin, rcsin b] (0, π 2 ] is strictly incresing, nd g = : [rcsin, rcsin b] (0, π f 2 ] [ b, ] stisfies g (x) = csc x( + 2 cot 2 x) > 0 (3.25) for x [rcsin, rcsin b].

6 96 CHU YUMING, ZHANG XIAOMING AND TANG XIAOMIN Eqution (3.25), Theorem A nd Theorem yield () rcsin L(, b) < < rcsin x dx b(l(, b) ) rcsin b + (b L(, b)) rcsin. (3.26) L(, b) Equtions (3.23) nd (3.24) follow from eqution (3.26). References [] T. Zgrj, On continuous convex or concve functions with respect to the logrithmic men, Act Univ. Crolin. Mth. Phys., 46 (2005), 3 0. [2] J. Mtkowski, Affine nd convex functions with respect to the logrithmic men, Colloq. Mth., 95 (2003), [3] C. E. M. Perce, J. Pečrić, Some theorems of Jensen type for generlized logrithmic mens, Rev. Roumine Mth. Pures Appl., 40 (995), [4] J. Sándor, On the identric nd logrithmic mens, Aequtiones Mth., 40 (990), [5] C. O. Imoru, The power men nd the logrithmic men, Internt. J. Mth. Mth. Soc., 5 (982), [6] K. B. Stolrsky, The power nd logrithmic mens, Amer. Mth. Monthly, 87 (980), [7] T. P. Lin, The power men nd the logrithmic men, Amer. Mth. Monthly, 8 (974), [8] B. C. Crlson, The logrithmic men, Amer. Mth. Monthly, 79 (972), [9] H. Alzer, On n integrl inequlity, Anl. Numér. Théor. Approx., 8 (989), (Received: October 5, 2007) (Revised: November 7, 2007) Chu Yuming nd Tng Xiomin Deprtment of Mthemtics Huzhou Techers College Huzhou 33000, P. R. Chin E mil: chuyuming@hutc.zj.cn E mil: txm@hutc.zj.cn Zhng Xioming Hining Rdio nd TV University Hining 34400, P. R. Chin E mil: zjzxm79@26.com