It is conjectured that for any real \(k \geq 0\) there exists a constant \(C_ k\) such that \[ I_ k(T) = \int^ T_ 0 \left | \zeta \left( {1 \over 2} + it \right) \right |^{2k} dt \sim C_ k T(\log T)^{k^ 2}. \] It is known that \(C_ 0=1\), \(C_ 1=1\) and \(C_ 2 = 1/2 \pi^ 2\), but it is not clear what \(C_ k\) should be in the general case. Assuming the Riemann Hypothesis (RH), \textit{J. B. Conrey} and \textit{A. Ghosh} [Mathematika 31, 159-161 (1984; Zbl 0542.10034)] proved that for any fixed \(k \geq 0\) \[ I_ k(T) \geq \bigl( c_ k + o(1) \bigr) T(\log T)^{k^ 2}, \] where \[ c_ k = {1 \over \Gamma (k^ 2+1)} \prod_ p \left\{ \left( 1-{1 \over p} \right)^{k^ 2} \sum^ \infty_{m = 0} \left( {k (k + 1) \cdots (k+m-1) \over m!} \right)^ 2p^{-m} \right\}, \] and observed that \(c_ 0 = C_ 0\) and \(c_ 1 = C_ 1\) (but not \(c_ 2 = C_ 2)\). Assuming RH the author proves in particular that for any fixed \(k \in (0,2)\) \[ I_ k (T) \leq \left( {2 \over (k^ 2 + 1)(2-k)}c_ k + o(1) \right) T(\log T)^{k^ 2} \] and observes that the factor in front of \(c_ k\) is 1 at \(k=0,1\) and bounded by \({27 \over 25}\) on [0,1]. The method of proof is motivated by ideas from the author's previous paper [J. Lond. Math. Soc., II. Ser. 24, 65-78 (1981; Zbl 0431.10024)] and the above work by Conrey-Ghosh, but the convexity arguments in the author's previous paper are replaced by inequalities involving derivatives.