New generalizations of the interpolation for a continuous Hölder’s inequality \[ \int_ Y \exp\Big(\int_ X \log f(x,y)d\mu (x)\Big)d\nu (y) \leq \exp\Big[\int_ X \log\Big(\int_ Y f(x,y)d\nu (y)\Big)d\mu(x)\Big],\eqno (*) \]
where \((X,\mu)\) and \((Y,\nu)\) are \(\sigma\)-finite measure spaces with positive measures \(\mu\), \(\nu\) and \(\mu(X)=1\), are derived and proved. Specifically, it is shown that if \(\mu(X)=\nu(Y)=1\) and \(f\in L^{1}(\mu\times\nu)\) satisfy the condition that \(\delta \leq f\) on \(X\times Y\) for some \(\delta > 0,\) then \(h=\exp[\int_ X \log(\int_ Y g(t,x,y)d\nu(y))d\mu(x)]\) is an interpolation for \((*)\). Several consequences of this result are given and also an example is included to convey another condition on \(f\) for the interpolation in the case \(0<\nu(Y)<\infty.\)