For \(\alpha>-1\) and \(1\le q<\infty\), denote by \(L_{\alpha}^q\) the space of complex valued measurable functions defined on \([0,\infty)\) for which \(\Vert f\Vert_{q,\alpha}:=\left(\int_0^{\infty}\vert f(x)\vert x^{2\alpha+1}dx\right)^{\frac1q}<\infty\). For \(\sigma>0\), denote by \(\mathcal{E}(\sigma,q,\alpha)\), the set of even entire functions of exponential type at most \(\sigma\) which belong to \(L_{\alpha}^q\). Consider also the sup norm \(\Vert f\Vert_C:=\sup\{\vert f(x)\vert ,\;x\in[0,\infty)\}\). The paper deals with the study of the optimal constants and the corresponding extremal functions which can appear in the following two inequalities:

(a)

\(\Vert f\Vert_C\le M \cdot \Vert f\Vert_{q,\alpha}\) and

(b)

\(\vert f(0)\vert \le D \cdot \Vert f\Vert_{q,\alpha}\), when \(f\in \mathcal{E}(\sigma,q,\alpha)\).

The main result says that, for \(\alpha\ge-\frac12\) and \(1\le q<\infty\), we have \(M(\alpha,q,\sigma)=D(\alpha,q,\sigma)\), where \(M(\alpha,q,\sigma)\) and \(D(\alpha,q,\sigma)\) are the optimal constants in the inequalities (a) and (b), respectively. In addition, the sets of corresponding extremal functions in (a) and (b) coincide and there is given a characterization of the extremal functions. For \(1<q<\infty\), the extremal function is unique (excepting a multiplicative factor). Moreover, the uniform norm of extremal functions for inequality (a) is attained only at the point \(x=0\).