To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function/functions $f$, for which we have the ("two-qubit separability") probability results $f(\frac{1+t}{2})=\frac{25}{341} =0.0733138$ and $f(\sqrt{t}) =1 -\frac{256}{27 \pi^2}=0.0393251$. (Also, at least in a limiting sense, $f(\frac{2 t}{1+t}) =0.$)

The possible arguments of $f$ in which we are interested are the members of the (infinite) class of operator monotone functions (of $t$) . (Theorem 7 of https://www.sciencedirect.com/science/article/pii/0024379594002118 tells us that such functions $g(t)$ satisfy the relation $g(t)=t g(t^{-1})$.)

Other—than the three already given ($\frac{1+t}{2}$ (the minimal), $\sqrt{t}$ and $\frac{2 t}{1+t}$ (the maximal))--members of the infinite class of operator monotone functions for which we have (2004) calculations (but only of a numerical nature, accurate to at most four decimal places, we believe) are for $f(\frac{t^{(t-1)}}{e}) \approx 0.0609965$, $f(\frac{1}{4} \left(\sqrt{t}+1\right)^2) \approx 0.0503391$ and $f(\frac{(t-1)}{\log{t}}) \approx .0346801$. (Table II, p. 14 in https://arxiv.org/abs/quant-ph/0308037), and also $f(\frac{1+6 t +t^2}{4 +4 t}) \approx 0.0475438$ (Table I there).

For background on the first value ($\frac{25}{341}$) of $f$ given, see https://arxiv.org/abs/1901.09889, and for the second ($1-\frac{256}{27 \pi^2}$), see eq. (87) in https://arxiv.org/abs/1701.01973 .

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit separability") probability results $f(\frac{1+t}{2})=\frac{25}{341} =0.0733138$ and $f(\sqrt{t}) =1 -\frac{256}{27 \pi^2}=0.0393251$. (Also, at least in a limiting sense, $f(\frac{2 t}{1+t}) =0.$)

The possible arguments of $f$ in which we are interested are the members of the (infinite) class of operator monotone functions (of $t$) . (Theorem 7 of https://www.sciencedirect.com/science/article/pii/0024379594002118 tells us that such functions $g(t)$ satisfy the relation $g(t)=t g(t^{-1})$.)

Other—than the three already given ($\frac{1+t}{2}$ (the minimal), $\sqrt{t}$ and $\frac{2 t}{1+t}$ (the maximal))--members of the infinite class of operator monotone functions for which we have (2004) calculations (but only of a numerical nature, accurate to at most four decimal places, we believe) are for $f(\frac{t^{(t-1)}}{e}) \approx 0.0609965$, $f(\frac{1}{4} \left(\sqrt{t}+1\right)^2) \approx 0.0503391$ and $f(\frac{(t-1)}{\log{t}}) \approx .0346801$. (Table II, p. 14 in https://arxiv.org/abs/quant-ph/0308037), and also $f(\frac{1+6 t +t^2}{4 +4 t}) \approx 0.0475438$ (Table I there).

For background on the first value ($\frac{25}{341}$) of $f$ given, see https://arxiv.org/abs/1901.09889, and for the second ($1-\frac{256}{27 \pi^2}$), see eq. (87) in https://arxiv.org/abs/1701.01973 .

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function/functions $f$, for which we have the ("two-qubit separability") probability results $f(\frac{1+t}{2})=\frac{25}{341} =0.0733138$ and $f(\sqrt{t}) =1 -\frac{256}{27 \pi^2}=0.0393251$. (Also, at least in a limiting sense, $f(\frac{2 t}{1+t}) =0.$)

The possible arguments of $f$ in which we are interested are the (infinite class of) operator monotone functions (of $t$) . (https://www.sciencedirect.com/science/article/pii/0024379594002118 Theorem 7 there tells us that such functions $g(t)$ satisfy the relation $g(t)=t g(t^{-1})$.)

Other—than the already given $\frac{1+t}{2}$ (the minimal), $\sqrt{t}$ and $\frac{2 t}{1+t}$ (the maximal)--members of the class of operator monotone functions for which we have (2004) calculations (but only of a numerical nature, accurate to at most four decimal places, we believe) are for $f(\frac{t^{(t-1)}}{e}) \approx 0.0609965$, $f(\frac{1}{4} \left(\sqrt{t}+1\right)^2) \approx 0.0503391$ and $f(\frac{(t-1)}{\log{t}}) \approx .0346801$. (Table II, p. 14 in https://arxiv.org/abs/quant-ph/0308037), and also $f(\frac{1+6 t +t^2}{4 +4 t}) \approx 0.0475438$ (Table I there).

For background on the first value ($\frac{25}{341}$) of $f$ given, see https://arxiv.org/abs/1901.09889, and for the second ($1-\frac{256}{27 \pi^2}$), see eq. (87) in https://arxiv.org/abs/1701.01973 .

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function/functions $f$, for which we have the ("two-qubit separability") probability results $f(\frac{1+t}{2})=\frac{25}{341} =0.0733138$ and $f(\sqrt{t}) =1 -\frac{256}{27 \pi^2}=0.0393251$. (Also, at least in a limiting sense, $f(\frac{2 t}{1+t}) =0.$)

The possible arguments of $f$ in which we are interested are the members of the (infinite) class of operator monotone functions (of $t$) . (Theorem 7 of https://www.sciencedirect.com/science/article/pii/0024379594002118 tells us that such functions $g(t)$ satisfy the relation $g(t)=t g(t^{-1})$.)

Other—than the three already given ($\frac{1+t}{2}$ (the minimal), $\sqrt{t}$ and $\frac{2 t}{1+t}$ (the maximal))--members of the infinite class of operator monotone functions for which we have (2004) calculations (but only of a numerical nature, accurate to at most four decimal places, we believe) are for $f(\frac{t^{(t-1)}}{e}) \approx 0.0609965$, $f(\frac{1}{4} \left(\sqrt{t}+1\right)^2) \approx 0.0503391$ and $f(\frac{(t-1)}{\log{t}}) \approx .0346801$. (Table II, p. 14 in https://arxiv.org/abs/quant-ph/0308037), and also $f(\frac{1+6 t +t^2}{4 +4 t}) \approx 0.0475438$ (Table I there).

For background on the first value ($\frac{25}{341}$) of $f$ given, see https://arxiv.org/abs/1901.09889, and for the second ($1-\frac{256}{27 \pi^2}$), see eq. (87) in https://arxiv.org/abs/1701.01973 .