Let \(c,A,B\in \mathbb R\) with \(A,B\ge 0\). If \((A-B)^2\le 8c^2(A+B-2c^2)\), then \(c\le (\sqrt A+\sqrt B)/2\).

Let \(a,b,t\in \mathbb R\) with \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(0\le t\le b\). Then \(a+t,a-t\in (0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,t,k\in \mathbb R\) and \(a\pm t\in (0,1)\), then the derivative of \(x\mapsto I(a+x)^2+kI(a-x)^2\) at \(t\) is \(-2I(a+t)Q(a+t)+2kI(a-t)Q(a-t)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,t,k\in \mathbb R\) and \(a\pm t\in (0,1)\), then the derivative at \(t\) of \(x\mapsto -2I(a+x)Q(a+x)+2kI(a-x)Q(a-x)\) is \(2(Q(a+t)^2+kQ(a-t)^2-1-k)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \(x\mapsto I(a+x)^2-I(a-x)^2\) is concave on \([0,b]\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \((I(a+b)^2-I(a-b)^2)^2\le 16I(a)^2Q(a)^2b^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in \mathbb R\) satisfy \(a-b\gt 0\) and \(a+b\lt 1\), then \(x\mapsto I(a+x)^2+I(a-x)^2+2(1-Q(a)^2)x^2\) is convex on \([-b,b]\). Here \(Q(p)=0\) and \(I(p)=\varphi (0)\) outside \((0,1)\), and \([-b,b]\) may be empty.

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), and \(a+b\lt 1\), then \(2Q(a)^2b^2\le I(a+b)^2+I(a-b)^2+2b^2-2I(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\) and \(a-b,a+b\in (0,1)\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in \mathbb R\) and \(a-b,a+b\in (0,1)\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

If \(a\in (0,1)\), \(a+b\in [0,1]\), and \(0\le \rho \lt 1\), then \(a+\rho b\in (0,1)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a,a+b\in [0,1]\), then \(\lim _{\rho \to 1^-}\overline I(a+\rho b)=\overline I(a+b)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a\in (0,1)\) and \(a+b,a-b\in [0,1]\), then \(\overline I(a)\le \bigl(\sqrt{\overline I(a+b)^2+b^2}+\sqrt{\overline I(a-b)^2+b^2}\bigr)/2\).