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Consider the complex-linear two-coordinate map $H(x,y)=(x+y,x-y)$. The triangle inequality shows that its operator norm from ${\ell }^1$ to ${\ell }^{\infty }$ is one; equality is attained at $(1,0)$. The complex parallelogram identityshows that its norm from ${\ell }^2$ to ${\ell }^2$ is $\sqrt{2}$. Interpolate these endpoint bounds by the Riesz-Thorin theorem with parameter $\eta =2/p$. ThenDividing the output norm by $2^{1/p}$ proves the scalar estimate:Apply it pointwise to $f,g$ and integrate its $p$th power. With $r=p/p^{\prime }=p-1>1$, this givesBy Minkowski inequality, the expression on the right is at mostTherefore the requested function inequality holds, namely the dual-exponent Clarkson inequality.