J. Korean Math. Soc. 47 (2010), No. 3, pp. 537–546 DOI 10.4134/JKMS.2010.47.3.537 DISCRETE MULTIPLE HILBERT TYPE INEQUALITY WITH NON-HOMOGENEOUS KERNEL Biserka Draˇ sˇ ci´ c Ban, Josip Peˇ cari´ c, Ivan Peri´ c, and Tibor Pog´ any Abstract. Multiple discrete Hilbert type inequalities are established in the case of non-homogeneous kernel function by means of Laplace integral representation of associated Dirichlet series. Using newly derived integral expressions for the Mordell-Tornheim Zeta function a set of subsequent special cases, interesting by themselves, are obtained as corollaries of the main inequality. 1. Introduction Let ℓ p be the space of all complex sequences x =(x n ) ∞ n=1 with the finite norm ‖x‖ p := ( ∑ ∞ n=1 |x n | p ) 1/p endowed. Let a =(a n ) ∞ n=1 ∈ ℓ p , b =(b n ) ∞ n=1 ∈ ℓ q be nonnegative sequences and 1/p +1/q =1,p> 1. Then (1)  m,n∈N a m b n m + n < π sin(π/p) ‖a‖ p ‖b‖ q , where constant π/ sin(π/p) is the best possible [3, p. 253]. This is the famous discrete Hilbert double series theorem or Hilbert inequality, a topic of interest of many mathematicians now-a-days too. The accustomary approach to deriving Hilbert’s inequality is by applying the H¨older inequality to suitably transformed Hilbert type double sum expression, i.e., to the bilinear form (2) H a,b K :=  m,n∈N K(m, n) a m b n , where a, b are nonnegative and K(·, ·) is the kernel function (of the double series (2)). Received August 1, 2008. 2000 Mathematics Subject Classification. Primary 26D15; Secondary 40B05, 40G99. Key words and phrases. discrete Hilbert type inequality, discrete multiple Hilbert type inequality, Dirichlet-series, non-homogeneous kernel, homogeneous kernel, multiple H¨older inequality, Tornheim’s double sum, Witten Zeta function, Mordell-Tornheim Zeta function. The authors were supported in parts by the Ministry of Sciences, Education and Sports of Croatia under Research Project No. 112-2352818-2814 (Draˇ sˇ ci´ c Ban & Pog´any), 117- 1170889-0888 (Peˇ cari´ c) and 058-1170889-1050 (Peri´ c). c 2010 The Korean Mathematical Society 537