A Hilbert-type Integral Inequality with Parameters

Transcription

1 Global Journal of Science Frontier Research Vol. Issue 6(Ver.) October P a g e 87 A Hilbert-type Integral Inequality with Parameters Peng Xiuying, Gao Mingzhe GJSFR- F Clasification FOR 8,6,33B,6D5 6D5 Abstract.-In this paper it is shown that a Hilbert-type integral inequality can be established by introducing two parameters ( ) λ (λ>). And the constant factor epressed by -function is proved to be the best possible. And then some important especial results are enumerated. As applications, some equivalent forms are given. Key words: Hilbert-type integral inequality, weight function function, Euler number,catalan constant. () Mathematical Subject Classification: 33B, 6D5,. ntroduction Lemmas Let,, f g L. Then f g y ddy f d g d y (.) where the constant factor is the best possible. And the equality contained in (.) holds if only if f( ), or g ( ). This is the famous Hilbert integral inequality,(see [],[]). Owing to the importance of the Hilbert inequality the Hilbert type inequality in analysis applications, some mathematicians have been studying them. Recently, various improvements etensions of (.) appear in a great deal of papers (see [3]-[]etc.). Specially, Gao Hsu enumerated the research articles more than 4 in the paper [6]. For convenience, we define establish the Hilbert-type integral inequality of the form y, when y. The purpose of the present paper is to y f g y ddy ( ) ( ) y C f d g d (.) where. We will give the constant factor C the epression of the weigh function ( ), prove the constant factor C to be the best possible, then give some important especial results, study some equivalent forms of them. Evidently, the inequality (.) is an etension of (.). The new inequality established is significant in theory applications. In order to prove our main results, we need the following lemmas. Lemma.. Let a be a positive number. Then a ( ) e d. a. (.3)

2 P a g e 88 Vol. Issue 6 (Ver.) October Global Journal of Science Frontier Research Proof. According to the definition of -function, we obtain immediately (.3). This result can be also found in the paper [] (pp. 6, formula 53). Lemma.. Let a be a positive number. Then G d (.4) cosh a a where G is Catalan constant, i.e. G Proof. Let. Eping the hyperbolic secant function, then using Lemma. we have cosha a e a k ( k) a d d e e d cosh a e a k k k (k) a ( ) ( ) e d k a k(k) ( ) G( ) a (.5) where the function G( ) is defined by i.e. G( ) ( ) k (.6) k (k ) Let.Then ( ). In accordance with the definition of the Catalan constant (see [], pp.53.), G k ( ) (k ) k we obtain from (.5) the equality (.4) at once. Lemma.3. Let. Then t t dt t ( ) G ( ). (.7) where the function G( ) is defined by (.6). Proof. Substitution t it is easy to deduce that

3 Global Journal of Science Frontier Research Vol. Issue 6(Ver.) October P a g e 89 e t t dt d d d t. e cosh By using (.5), the equality (.7) follows. Lemma.4. With the assumptions as Lemma.3, then e e t t dt ( ) G( ) t (.8) where G( ) is defined by (.6). Proof. It is easy to deduce that t t t t t t t dt t dt t dt t t v t dt v v dv t t v v t dt v dv t t t dt. By Lemma.3, the equality (.8) follows at once.. Main results In this section, we will prove our assertions by using the above Lemmas. Theorem.. Let f g be two real functions,. If ( ) g d, then y f g y ddy C f d g y d ( ), (.) where the constant factor C is defined by f d C ( ) G( ) (.)

4 P a g e 9 Vol. Issue 6 (Ver.) October Global Journal of Science Frontier Research the function G( ) is defined by (.6) () z is -function. And the constant factor C in (.3) is the best possible. Proof. We can apply the Cauchy inequality to estimate the left-h side of (.) as follows. y f ( ) g( y) y where ( ) y y y ddy ( ) f ( ) d ( ) g ( ) d, (.3) dy. By proper substitution of variable, then by Lemma.4, it is easy to deduce that ( ) y y y dy t t du C (.4) where the constant factor C is defined by (.8). It follows from (.3) (.4) that ddy C f d g y d y f g y, (.5) If (.5) takes the form of the equality, then there eist a pair of non-zero constants c c such that y y y y c f c g y a.e. on (, ) (, ) Then we have y y c f c y g y C. (constant) a.e. on (, ) (, ) Without losing the generality, we suppose that c, then C c f d d. This contradicts that f d. Hence it is impossible to take the equality in (.5). So the inequality (.) is valid. It remains to need only to show that C in (.) is the best possible.. Define two functions by (, ) f g y It is easy to deduce that [, ) y (, ) y y[, )

5 Global Journal of Science Frontier Research Vol. Issue 6(Ver.) October P a g e 9 f d y g y dy. If C in (.) is not the best possible, then there eists K, such that K C H(, ) y f g y ddy y K f d y g ydy. K K f d y g ydy. (.6) On the other h, we have H (, ) y f g y y ddy y y y ddy y y dy d y t dt d t t t t dt t d dt t d d t t t dt t dt d t t t dt dt t t. (.7) When is sufficiently small, we obtain from (.7) that H t t (, ) () () t dt t dt

6 P a g e 9 Vol. Issue 6 (Ver.) October Global Journal of Science Frontier Research By (.4), we have t dt () t ( ) H(, ) C (). ( ) (.8) Evidently, the inequality (.8) is in contradiction with (.6). Therefore, the constant factor C in (.) is the best possible. Thus the proof of Theorem is completed. Based on Theorem., we have the following important results. Theorem.. If f ( ) d g ( ) d, then y f g y y ddy G f d g d (.9) 8. where G is the Catalan constant. And the constant factor 8G in (.9) is the best possible. Theorem.3. Let n be a nonnegative integer. If ( ) g d, then ( ) n ddy E y n f d g d f d n f g y y, (.) where E the E are the Euler numbers,viz. E, E 5, E3 6, E4 385, etc.. And the ns ' constant factor n E n in (.) is the best possible. Proof. We need only to show that the constant factor in (.) is true. When n, it is known from (.) that n C ( ) G( ) (n ) G( n) n k n. n k (k ) ( )! According to the paper [3] (pp. 3.), we have k n ( ) En. n n k (k) ( n)!

7 Global Journal of Science Frontier Research Vol. Issue 6 (Ver.), October P a g e 93 where the E are the Euler numbers,viz. E, E 5, E3 6, E4 385, etc. Notice that ns ' k ( ), hence we can define E. Whence we obtain (k ) 4 k n C E n. In particular, for case, based on Theorem.3, we get the following result at once. Theorem.4. g ( ) Let n be a nonnegative integer. If d, then f ( ) d n y f g y y ddy n E n f d g d, (.) where E the E ns ' are the Euler numbers. And the constant factor in (.) is the best n E n possible. Specially, when n, the inequality (.) follows from (.) immediately. Similarly, we can establish also a great deal of new inequalities. They are omitted here. 3. Some Equivalent Forms As applications, we will build some new inequalities. Theorem 3.. Let f be a real function,. If y ( ) y y f d dy C f d, (3.) f d, then where C is defined by (.) the constant factor C in (3.) is the best possible. And the inequality (3.) is equivalent to (.). Proof. First, we assume that the inequality (.) is valid. Setting a real function g( y) as By using (.), we have ( ) y y, g y y f d y(, ) y y y y y f d dy f g( y) ddy C f ( ) d y g ( y) dy y C f ( ) d y f ( ) d dy y (3.)

8 P a g e 95 Vol. Issue 6 (Ver.) October Global Journal of Science Frontier Research It follows from (3.) that the inequality (3.) is valid after some simplifications. On the other h, assume that the inequality (3.) keeps valid, by applying in turn the Cauchy inequality (3.), we have y y y f g yddy y f dy g ydy y y y y f d dy y g y dy C f d y g ydy C f d y g ydy. (3.3) Therefore the inequality (3.) is equivalent to (.). If the constant factor C in (3.) is not the best possible, then it is known from (3.3) that the constant factor C in (.) is also not the best possible. This is a contradiction. Theorem is proved. Theorem 3.. Let f be a real function. If y y 8 f ( ) f d dy G f d d, then (3.4) where G is the Catalan constant the constant factor (8 G) in (3.4) is the best possible. And the inequality (3.4) is equivalent to (.9). Its proof is similar to one of Theorem 3.. Hence it is omitted. Theorem 3.3. Let f be a real function, n N. If ( ) f d, then n n y y y f d dy E n f d, (3.5) where E the E are the Euler numbers,viz. E, E 5, E3 6, E4 385, etc.. And the ns ' constant factor n E n in (3.5) is the best possible. And the inequality (3.5) is equivalent to (.). Similarly, we can establish also some new inequalities. They are omitted here

9 Global Journal of Science Frontier Research Vol. Issue 6(Ver.) October P a g e 96 References [] G. H. Hardy, J. E. Littlewood G. Polya, Inequalities [M]. Cambridge: Cambridge Univ. Press, 95. [] Kuang Jichang, Applied Inequalities, 3nd. ed., Jinan, Shong Science Technology Press, 4, [3] He Leping, Jia Weijian Gao Mingzhe, A Hardy-Hilbert s Type Inequality With Gamma Function And Its Applications, Integral Transform Special Functions, Vol. 7, 5(6), [4] Gao Mingzhe, On Hilbert s Inequality Its Applications, J. Math. Anal. Appl. Vol., (997), [5] Gao Mingzhe, Tan Li L. Debnath, Some Improvements on Hilbert s Integral Inequality, J. Math. Anal. Appl., Vol. 9, (999), [6] Gao Mingzhe Hsu Lizhi, A Survey of Various Refinements And Generalizations of Hilbert s Inequalities, J. Math. Res. & Ep.,Vol. 5, (5), [7] Yang Bicheng, On a Basic Hilbert-type Integral Inequality Etensions, College Mathematics, Vol. 4, (8), [8] Hong Yong, All-sided Generalization about Hardy- Hilbert s Integral Inequalities, Acta Mathematica Sinica, Vol. 44, 4(), [9] Hu Ke, On Hilbert s Inequality, Chin.Ann. Math., Ser. B, Vol. 3, (99), [] He Leping, Gao Mingzhe Zhou Yu, On New Etensions of Hilbert s Integral Inequality, Int.. J. Math. & Math. Sci., Vol. 8 (8), Article ID 9758, -8. [] Yang Bicheng, A New Hilbert Type Integral Inequality Its Generalization, J. Jilin Univ. (Sci. Ed.), Vol. 43, 5(5), [] Jin Yuming, Table of Applied Integrals, Hefei, University of Science Technology of China Press, 6. [3] Wang Lianiang Fang Dehi, Mathematical Hbook, Beijing, People s Education Press, 979, 3. 9