ON THE HILBERT INEQUALITY. 1. Introduction. π 2 a m + n. is called the Hilbert inequality for double series, where n=1.

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1 Acta Math. Univ. Comenianae Vol. LXXVII, 8, pp ON THE HILBERT INEQUALITY ZHOU YU and GAO MINGZHE Abstract. In this paper it is shown that the Hilbert inequality for double series can be improved by introducing a weight function of the form n n n+ n+ ln n, π where n N. A similar result for the Hilbert integral inequality is also given. As applications, some sharp results of Hardy-Littlewood s theorem and Widder s theorem are obtained.. Introduction Let {a n } and {b n } be two sequences of comple numbers. It is all-round known that the inequality a m bn. π a m + n n b n m is called the Hilbert inequality for double series, where a n < + and b n < +, and that the constant factor π in. is the best possible. The equality in. holds if and only if {a n }, or {b n } is a zero-sequence see [?]. The corresponding integral form of. is that fgy + y ddy. π f d g d where f d < + and g d < +, and that the constant factor π in. is also the best possible. The equality in. holds if and only if f, or g. Recently, various improvements and etensions of. and. appeared in a great deal of papers see [?]. The purpose of the present paper is to build the Hilbert inequality with the weights by means of a monotonic function Received August 3, 7; revised February, 8. Mathematics Subject Classification. Primary 6D5. Key words and phrases. Hilbert s inequality; weight function; double series; monotonic function; Hardy-Littlewood s theorem; Widder s theorem. A Project Supported by Scientific Research Fund of Hunan Provincial Education Department 6C657.

2 36 ZHOU YU and GAO MINGZHE of the form +, thereby new refinements of. and. are established, and then to give some of their important applications. For convenience, we need the following lemmas..3 Lemma.. Let n N. Then d n + + n + Proof. Let a, e and f be real numbers. Then d a + e + f e + a f π n + ln n { f ln e + f lna + + e a arctan a } + C where C is an arbitrary constant. This result has been given in the papers see [3] [4]. Based on this indefinite integral it is easy to deduce that the equality.3 is true. Lemma.. Let n N,, +. Define two functions by n f + n + + n n g + + n + +, n then f and g are monotonously decreasing in, +, and.4.5 f d π πω n g d π + πω n where the weight function ω is defined by n n ω n ln n.6 n + n + π Proof. At first, notice that f f + f, where f + n + n + +, hence we can write f in form, f n + n +. It is obvious that f and f are monotonously decreasing in, +. Hence f is monotonously decreasing in, +. Net, notice that n + n + n,

3 ON THE HILBERT INEQUALITY 37 we can write g in form g g + g, where g n n + n + n, g + n +. It is obvious that g and g are monotonously decreasing in, +. Hence g is also monotonously decreasing in, +. Further we need only to compute two integrals. f d + n n + + n + n + + n π n π π n n + n d + n π + n n + t dt + d n + t + t dt + n + d + d n + t + t dt n π + n n π + n By Lemma., we obtain.7 f d π { π π n + + n ln n n + } n π + n The equality.4 follows from.7 at once after some simple computations and simplifications. Similarly, the equality.5 can be obtained.. Main Results First, we establish a new refinement of..

4 38 ZHOU YU and GAO MINGZHE Theorem.. Let {a n } and {b n } be two sequences of comple numbers. If a n < + and b n < +, then. m a m bn m + n 4 π 4 a n ω n a n b n ω n b n where the weight function ω n is defined by.6. Proof. Let c be a real function and satisfy the condition c n+c m, n, m N. Firstly we suppose that b n a n. Applying Cauchy s inequality we have a m ā n a m ā n c n + c m. where m m + n m m m + n a m cn + cm/ m m + n/ n a n cn + cm/ m + n/ m J J J a m m m+n n cn + cm m J cn + cm m m ā n m+n We can write the double series J in the following form: J cm + cn a n. m + n m m m /4 /4 Let c +. It is obvious that + + n +. It is known from n Lemma. that the function f is monotonously decreasing. Hence we have m n J m + n m + m + + a n n + n π a n π ω n a n + n + n d a n

5 ON THE HILBERT INEQUALITY 39 where the weight function ω n is defined by.6. Similarly, n J + + n + + d n ā n π a n + π ω n a n. { Whence J J π a n Consequently, we have a m ā n.3 m + n m ω n a n }. π a n ω n a n where the weight function ω n is defined by.6. If b n a n, then we can apply Schwarz s inequality to estimate the right-hand side of. as follows: 4 a m bn m + n a m t m bn t n dt.4 m m a m t m dt b n t n dt m a m ā n b m bn m + n m + n m m And then by using the relation.3, from.4 and the inequality., we obtain at once. Similarly, we can establish a new refinement of.. Theorem.. Let f and g be two functions in comple number field. If f d < +, g d < +, then.5 fg + y ddy 4 π 4 f d g d ω f d ω g d

6 3 ZHOU YU and GAO MINGZHE where the weight function ω is defined by ω ln.6 > + + π Its proof is similar to that of Theorem., it is omitted here. For the convenience of the applications, we list the following result. Corollary.3. Let f be a function in comple number field. If f d < +, then ff y ddy + y.7 π f d ω f d where the weight function ω is defined by Applications As applications, we shall give some new refinements of Hardy-Littlewood s theorem and Widder s theorem. Let f L, and f for all. Define a sequence {a n } by a n n fd, n,,,.... Hardy-Littlewood [] proved that 3. a n < π n f d, where π is the best constant that the inequality 3. keeps valid. Theorem 3.. Let f L, and f for all. Define a sequence {a n } by a n n / fd n,,.... Then 3. an π a n ω n a n f d where ωn is defined by.6. Proof. By our assumptions, we may write a n in the form a n a n n / fd.

7 ON THE HILBERT INEQUALITY 3 Applying Cauchy-Schwarz s inequality we estimate the right hand side of 3. as follows an a n n / fd a n n / fd 3.3 a n n / d m m a m a n m+n d a m a n m + n f d f d f d It is known from.3 and 3.3 that the inequality 3. is valid. Therefore the theorem is proved. Let a n n,,,...., A a n n, A 3.4 A d π This is Widder s theorem see []. n e A d n a n n n!. Then Theorem 3.. With the assumptions as the above-mentioned, it yields 3.5 A d π e A d where ω is defined by.6. Proof. At first we have the following relation: e t A tdt e t n a n n n n! a n t n dt n! t n e t dt ω e A d a n n A n

8 3 ZHOU YU and GAO MINGZHE Let t s. Then we have A d e t A t dt d 3.6 e sy A sds dy e su f s ds du i e s A s ds d e su+ A s ds du f s f t dsdt s + t where f e A. By Corollary.3, the inequality 3.5 follows from 3.6 at once. References. Hardy G. H., Littlewood J. E. and Polya G., Inequalities, Cambridge Univ. Press, Cambridge, U.K., 95.. Gao Mingzhe and Hsu Lizhi, A survey of various refinements and generalizations of Hilbert s inequalities, Journal of Mathematical Research and Eposition 5 5, Zwillinger D. et al., CRC Standard Mathematical Tables and Formulae, CRC Press, Gradshteyn I. S. and Ryzhik I. M., Table of Integrals, Series, and Products, Academic Press,. Zhou Yu, Department of Mathematics and Computer Science Normal College, Jishou University Jishou Hunan 46, P. R. China, hong99@63.com Gao Mingzhe, Department of Mathematics and Computer Science Normal College, Jishou University Jishou Hunan 46, P. R. China, mingzhegao@63.com