Let \(1\lt p\lt 2\) and \(0\le t\lt 1\). Then \((1+t)^{p-1}-(1-t)^{p-1}\le (p-1)t((1+t)^{p-2}+(1-t)^{p-2})\).

Let \(1\lt p\lt 2\). The function \(z\mapsto (1+\sqrt z)^p+(1-\sqrt z)^p\) is convex on \([0,1]\).

Let \(1\lt p\lt 2\). The function \(z\mapsto ((1+\sqrt z)^p+(1-\sqrt z)^p)^{2/p}\) is convex on \([0,1]\).

Let \(1\lt p\lt 2\), \(0\lt t\lt 1\), and \(0\le z\le 1\). Put \(S=(1+t)^p+(1-t)^p\). Then \(S^{2/p}+S^{2/p-1}\frac{(1+t)^{p-1}-(1-t)^{p-1}}t(z-t^2)\le ((1+\sqrt z)^p+(1-\sqrt z)^p)^{2/p}\).

Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), and \(\alpha ^p+\beta ^p=1\). Put \(t=(\beta -\alpha )/(\beta +\alpha )\) and \(S=(1+t)^p+(1-t)^p\). Then \(S^{2/p}=4/(\alpha +\beta )^2\) and \(S^{2/p-1}((1+t)^{p-1}-(1-t)^{p-1})/t=2(\beta ^{p-1}-\alpha ^{p-1})/(\beta -\alpha )\).

Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), \(\alpha ^p+\beta ^p=1\), and \(-1\le y\le 1\). Put \(r=(\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)\). Setting \(u=(1+y)/\alpha \) and \(v=(1-y)/\beta \), one has \((\alpha ^pu+\beta ^pv)^2+r\alpha ^p\beta ^p(u-v)^2\le ((1+y)^p+(1-y)^p)^{2/p}\).

Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), \(\alpha ^p+\beta ^p=1\), and \(u,v\ge 0\). Put \(r=(\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)\). Then \((\alpha ^pu+\beta ^pv)^2+r\alpha ^p\beta ^p(u-v)^2\le (\alpha ^pu^p+\beta ^pv^p)^{2/p}\).