We generalize several inequalities involving powers of the numerical radius for the product of two operators acting on a Hilbert space. Moreover, we give a Jensen operator inequality for strongly convex functions. As a corollary, we improve the operator Hölder-McCarthy inequality under suitable conditions. In particular, we prove that if $f:J\rightarrow \mathbb{R}$ is strongly convex with modulus $c$ and differentiable on ${\rm int}(J)$ whose derivative is continuous on ${\rm int}(J)$ and if $T$ is a self-adjoint operator on the Hilbert space $\cal{H}$ with $\sigma(T)\subset {\rm int}(J)$, then $$\langle T^2x,x\rangle-\langle Tx,x\rangle^2\leq \dfrac{1}{2c}(\langle f'(T)Tx,x\rangle -\langle Tx,x\rangle \langle f'(T)x,x\rangle)$$ for each $x\in\cal{H}$, with $\|x\|=1$.