Research Article A New Method to Study Analytic Inequalities

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1 Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 200, Article ID 69802, 3 ages doi:0.55/200/69802 Research Article A New Method to Study Analytic Inequalities Xiao-Ming Zhang and Yu-Ming Chu Deartment of Mathematics, Huzhou Teachers College, Huzhou 33000, China Corresondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn Received 6 October 2009; Acceted 24 December 2009 Academic Editor: Kunquan Lan Coyright q 200 X.-M. Zhang and Y.-M. Chu. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original wor is roerly cited. We resent a new method to study analytic inequalities involving n variables. Regarding its alications, we roved some well-nown inequalities and imroved Carleman s inequality.. Monotonicity Theorems Throughout this aer, we denote R the set of real numbers and R the set of strictly ositive real numbers, n N, n 2. In this section, we resent the main results of this aer. Theorem.. Suose that a, b R with a<band c a, b, f : a, b n artial derivatives and R has continuous D m x,x 2,...,x n,c min c, x m max / c n n, m, 2,...,n.. If f x / x m > 0 for all x D m m, 2,...,n,then f ( y,y 2,...,y n,c f c,c,...c,c,.2 for all y m c, b m, 2,...,n.

2 2 Journal of Inequalities and Alications Proof. Without loss of generality, since we assume that n 3andy >y 2 >c. For x y 2,y, we clearly see that x,y 2,c D, then f x x x x,y 2,c > 0..3 From the continuity of the artial derivatives of f and f x x x y2,y 2,c > 0,.4 we now that there exists ε>0 such that y 2 ε c and f x x x x,y 2,c > 0,.5 for any x y 2 ε, y 2. Hence, since f,y 2,c : x y 2 ε, y f x,y 2,c is strictly monotone increasing, then we have f ( y,y 2,c >f ( y 2,y 2,c >f ( y 2 ε, y 2,c..6 Next, for x 2 y 2 ε, y 2, then y 2 ε, x 2,c D 2 and f x x 2 x y2 ε,x 2,c > 0..7 Hence, we get f ( y,y 2,c >f ( y 2,y 2,c >f ( y 2 ε, y 2,c >f ( y 2 ε, y 2 ε, c..8 If y 2 ε c, then Theorem. is true. Otherwise, we reeat the above rocess and we clearly see that the first and second variables in f are decreasing and no less than c. Lets, t be their limit values, resectively, then f y,y 2,c >f s, t, c and s, t c. Ifs c, t c, then Theorem. is also true; otherwise, we reeat the above rocess again and denote and q the greatest lower bounds for the first and the second variables, resectively. We clearly see that q c; therefore, f y,y 2,c >f c, c, c and Theorem. is true. Similarly, we have the following theorem.

3 Journal of Inequalities and Alications 3 Theorem.2. Suose that a, b R with a<band c a, b, f : a, b n artial derivatives and R has continuous E m x,x 2,...,x n,c max c, x m min / c n n, m, 2,...,n..9 If f x / x m < 0 for all x E m m, 2,...,n,then f ( y,y 2,...,y n,c f c,c,...c,c,.0 for all y m a, c m, 2,...,n. It follows from Theorems. and.2 that we get the following Corollaries.3.6. Corollary.3. Suose that a, b R with a<b, f : a, b n R has continuous artial derivatives and D m x x,x 2,...,x n a min x <x m max x b, m, 2,...,n.. n n If f x / x m > 0 for all x D m and m, 2,...,n,then f x,x 2,...,x n f x min,x min,...,x min,.2 for all x m a, b m, 2,...,n with x min min n x. Corollary.4. Suose a, b R with a < b,then D x x,x 2,...,x n a min <x max b n n,.3 and f : a, b n R is symmetric with continuous artial derivatives. If f x / x > 0 for all x x,x 2,...,x n D,then f x,x 2,...,x n f x min,x min,...,x min,.4 where x min min n x. Equality holds if and only if x x 2 x n. Corollary.5. Suose a, b R with a<b, f : a, b n R has continuous artial derivatives and E m x x,x 2,...,x n a x m min x < max x b..5 n n

4 4 Journal of Inequalities and Alications If f x / x m < 0 for all x E m and m, 2,...,n,then f x,x 2,...,x n f x max,x max,...,x max,.6 where x max max n x. Equality holds if and only if x x 2 x n. Corollary.6. Suose a, b R with a < b,then E n x x,x 2,...,x n a x n min x < max x b,.7 n n and f : a, b n R is symmetric with continuous artial derivatives. If f x / x n < 0 for all x x,x 2,...,x n E n,then f x,x 2,...,x n f x max,x max,...,x max,.8 where x max max n x. Equality holds if and only if x x 2 x n. 2. Unifying Proof of Some Well-Known Inequality In this section, we denote a a,a 2,...,a n, a min min n a, a max max n a, and D m a a m a max >a min > 0, m, 2,...,n. 2. Proosition 2. Power Mean Inequality. If the ower mean M r a of order r is defined by M r a /n n i ar i /r for r / 0 and M 0 a n i a/n i,thenm r a M s a for r > s; equality holds if and only if a a 2 a n. Proof. It is well nown that M r a is symmetric with resect to a,a 2,...,a n and r M r a is continuous. Without loss of generality, we assume that r, s / 0. Then f a r ln ( n i ar i n ( n s ln i as i, a R n n, f a ar a n i ar i as n i as i n i 2( a r a s i as n i ar i n i as i a r i n i 2 as a r i [ a /a i r s ]. n i ar i n i as i 2.2

5 Journal of Inequalities and Alications 5 If a D, then f a / a > 0. It follows from Corollary.4 that we get f a,a 2,...,a n f a min,a min,...,a min, ( n /r i ar i n ( n i as i n /s, M r a M s a. 2.3 Equality holds if and only if a a 2 a n. Proosition 2.2 Holder Inequality. Suose that x,x 2,...,x n, y,y 2,...,y n R n, q >.If/ /q, then ( x / ( /q y q x y. 2.4 Proof. Let b b,b 2,...,b n R n and ( /q f : a R n /( b a b a b /q, a R n. 2.5 If a D, then f a ( /q a q b /( b a b q b a /q ( / ( / ( / q b a / b a b a / b a 2.6 > ( / ( / ( / q b a / b a b a / b a 0. Similarly, if a D m m 2, 3,...,n, then f a / a m > 0. From Theorem., weget f a,a 2,...,a n f a min,a min,...,a min, ( /q /( b a b a b /q. 2.7 Therefore, Proosition 2.2 follows from a y q /x and b x.

6 6 Journal of Inequalities and Alications Proosition 2.3 Minowsi Inequality. Suose that x,x 2,...,x n, y,y 2,...,y n R n.if >, then ( / ( / ( / x y ( x y. 2.8 Proof. Let b b,b 2,...,b n R n and If a D, then ( / ( ( / f : a R n b a b a /, a R n. 2.9 f a ( / a b b a ( b ( ( b a / a / ( a / ( b a b / ( / b a / ( / a / ( ( b / a / ( / b a ( ( / b a / ( b a b / ( ( / ( b a / ( / b a / a / a / 2.0 > ( ( / b a / ( b a b / ( ( / ( b a / ( b a / a / 0. a / / Similarly, If a D m m 2, 3,...,n, then f a / a m > 0. It follows from Theorem. that we get f a,a 2,...,a n f a min,a min,...,a min, ( / ( ( / ( b a b a / /. b 2. Therefore, Proosition 2.3 follows from a y /x and b x.

7 Journal of Inequalities and Alications 7 3. A Brief Proof for Hardy s Inequality If a n 0 n N, n with n a n <, then the well-nown Hardy s inequality see, Theorem 326 is ( ( a n a. n n n 3. In this section, we establish the following result involving Hardy s inequality. Theorem 3.. Let n N, n, and a 0 N,. If ( B n min / a, 3.2 n 2 then ( a a j j ( B n /2 ( /. j /2 j 3.3 Proof. Let b /2 / a, then inequality 3.3 is equivalent to ( b /2 b j ( / j /2 j ( B n /2 ( /, j /2 j 3.4 and B n min n b.let D m b b m max b > min b > 0, m, 2,...,n, n n ( f : b 0, n b /2 b j ( /. j /2 j 3.5

8 8 Journal of Inequalities and Alications If b D m m, 2,...,n, then f b b m ( b m m /2 m b ( m > m /2 / m /2 / b j m /2 / ( / j /2 j m ( /. j /2 j 3.6 Maing use of the well-nown Hadamard s inequality of convex functions, we get f b b m > b ( m m /2 / b m m /2 / b m > m /2 / m /2 / [ ( m ( /2 /2 m x /2 ( ] m /2 2 / / / dx [ ( ( m /2 x 2 / dx / m /2 ] Then Theorem. leads to f b,b 2,...,b n f B n,b n,...,b n, 3.8 and we clearly see that inequalities 3.4 and 3.3 are true. Corollary 3.2. Let n N, n, and a 0 N,. If ( B n min / a, 3.9 n 2 then ( a a j j ( ( >B n 2 B n. 3.0

9 Journal of Inequalities and Alications 9 Proof. From inequality 3.3, we clearly see that ( b /2 b j ( / j /2 j [ ( ( >B n /2 ( ( B n /2 ( B n 2 ( B n. /2 /2 x /2 / dx ] 3. Remar 3.3. If n, then inequality 3. follows from inequality A Refinement of Carleman s Inequality If a n 0 n N, n with 0 < n a n <, then the well-nown Carleman s inequality is ( n /n a <e a n, n n 4. with the best ossible constant factor e see 2. Recently, Yang and Debnath 3 gave a strengthened version of 4. as follows: ( n /n ( a <e a n. 2n 2 n n 4.2 Some other strengthened versions of 4. were given in 4 9. In this section, we give a refinement for Carleman s inequality see Corollary 4.4. Lemma 4.. If m N and m, then ( 2 e 3m 7 m > m!, / ( 2 e 3m 0 m > m!. / m

10 0 Journal of Inequalities and Alications Proof. Let ψ m e 2/ 3m 7 /m m /! /, then inequality ψ m >ψ m is equivalent to inequality 2m 2 3m 7 2m 3m 0 > m e m!. /m 4.5 If m 6, then simle comutation leads to inequality 4.5. If m 7, then it is not difficult to verify that 2πm e 7/3 and 2πm e 2m 2 7m 70 / 9m 2 39m If x>0, then e> /x x ;thisimliesthat ( e> 2m2 7m 70 9m 2 39m 50 m 9m 2 39m 50 m/ 2m 2 7m From inequalities 4.6 and 4.7, weget m 2πm > ( 2m2 7m 70, 9m 2 39m 50 m 2πm / 2m m 3m 7 3m 0 >, m 9m 2 39m 50 m 5 3m 7 2m 3m 0 > m m 2πm. / 2m 4.8 From the well-nown Stirling Formula m! 2πm m/e m ex θ m /2m 0 <θ m <, we get m! > ( m m 2πm. 4.9 e Therefore, inequality 4.5 follows from inequalities 4.8 and 4.9. From the monotonicity of sequence ψ m m and lim m ψ m 0, we get ψ m > 0; therefore, inequality 4.3 is roved.

11 Journal of Inequalities and Alications Meanwhile, we have 2π m >e 2/3, 2π m >e 2m 2 / 3m 8, ( 2 3m 8 /2 2m 2 / 3m 8 2π m >, 3m 8 ( e 2π m / 2m 2 > 2 3m 0 m > 3m 0 3m 8, e m 2π m / 2m Therefore, inequality 4.4 follows from inequalities 4.0 and ( m m m! > 2π m. 4. e Theorem 4.2. Let n N, n, and a > 0, 2,...,n.IfB n min n a,then ( 2 e a 3 7 a j j / ( ] 2 B n [e ! / Proof. Let b a,, 2,...,n,andb b,b 2,...,b n, D m b b m max b > min b > 0, m, 2,...,n, n n ( f : b R n 2 b e 3 7 b j! j /, b R n. 4.3 Then inequality 4.2 is equivalent to the following inequality: ( 2 b e 3 7 b j! j / ( ] 2 B n [e 3 7, 4.4! / where B n min n b.

12 2 Journal of Inequalities and Alications If b D m m, 2,...,n, then f b b m ( 2 e 3m 7 m m ( 2 >e 3m 7 m m ( 2 >e 3m 7 m m b j b m!! /!. / j / 4.5 From inequality 4.3 and f b / b m > 0 together with Theorem., we clearly see that f b,b 2,...,b n f B n,b n,...,b n. 4.6 Therefore, inequality 4.4 is roved. Corollary 4.3. Let n N, n, and a > 0, 2,...,n. IfB n min n a,then ( 2 e a 3 7 a j j / B n ( 4 5 e. 4.7 Proof. Let T m e m 2/ 3 7 / m /! / m, 2,...,n, then inequality 4.4 imlies that T m n m is a strictly increasing sequence. Then from inequality 4.2 we get ( 2 e a 3 7 a j j / B n T n B n T B n ( 4 5 e. 4.8 Let n ; thus, we now that Corollary 4.4 is true. Corollary 4.4. If a n 0 n N,n with 0 < n a n <,then n a j n j /n ( 2 e a n. 3n 7 n 4.9 Remar 4.5. Many other alications for Theorem. aeared in 0. Acnowledgments The authors wish to than the anonymous referees for their very careful reading of the manuscrit and fruitful comments and suggestions. This wor was artly suorted by the

13 Journal of Inequalities and Alications 3 National Nature Science Foundation of China under Grant no , the Nature Science Foundation of Zhejiang Province under Grant no. Y60728, the Nature Science Foundation of China Central Radio & TV University under Grant no. GEQ633, and the Nature Science Foundation of Zhejiang Broadcast & TV University under Grant no. XKT-07G9. References G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, T. Carleman, Sur les fonctions quasi-analytiques, in Comtes rendus du Ve Congres des Mathematiciens, Scandinaves,. 8 96, Helsini, Finland, B. Yang and L. Debnath, Some inequalities involving the constant e, and an alication to Carleman s inequality, Journal of Mathematical Analysis and Alications, vol. 223, no., , H. Alzer, On Carleman s inequality, Portugaliae Mathematica, vol. 50, no. 3, , H. Alzer, A refinement of Carleman s inequality, Journal of Aroximation Theory, vol. 95, no. 3, , M. Johansson, L.-E. Persson, and A. Wedestig, Carleman s inequality-history, roofs and some new generalizations, JIPAM Journal of Inequalities in Pure and Alied Mathematics, vol. 4, no. 3, article 53, 9 ages, H. P. Liu and L. Zhu, New strengthened Carleman s inequality and Hardy s inequality, Journal of Inequalities and Alications, vol. 2007, Article ID 8404, 7 ages, J. Pečarić and K. B. Stolarsy, Carleman s inequality: history and new generalizations, Aequationes Mathematicae, vol. 6, no. -2, , G. I. Sunouchi and N. Taagi, A generalization of the Carleman s inequality theorem, Proceedings of the Physico-Mathematical Society of Jaan, vol. 6, , X. M. Zhang and Y. M. Chu, New Discussion to Analytic Inequality, Harbin Institute of Technology Press, Harbin, China, 2009.