23 problems

Let $E(r)$ be the complete elliptic integral of the second kind, let $r^{\prime}=\sqrt{1-r^{2}}$, and let $A_{p}(a,b)=\left((a^{p}+b^{p})/2\right)^{1/p}$ for $p\neq 0$, with…

Both-types invariance conjecture. The only such pairs are the quasi-arithmetic transforms

Sufficient inequalities for Stolarsky's bounds. The inequalities

Stolarsky's bounds conjecture. For all $(a,b,c)\in R_{+}^{3}$,

Residuality conjecture. Residuality with exponent $\alpha\in(2,3)$ refers to the Hölder continuity of the second derivative.

Sign conjecture. Then

Sign conjecture. At $\lvert\alpha\rvert=\frac12\sqrt{2\pm\sqrt3}$,

Sign conjecture. The inequality

Let $K$, $M$, and $N$ be means, and let $G$ denote the geometric mean. For means $A$ and $B$, write $A\otimes B$ for their indicated mean composition. Uniqueness conjecture for sta…

Let $M$ be a symmetric homogeneous mean that has an asymptotic power series expansion, and suppose that it fulfills the requirements of the open question from Farhi. Let $L_c$ deno…

Local Hardy characterization conjecture. The quasi-arithmetic mean $\mathscr{A}^{[f]}$ has the weak-Hardy property if and only if there exists $\varepsilon\in I$ such that…

Hindman–Ellis conjecture. Every compact convex subsystem of $M(S)$ contains an idempotent mean.

Exponent-range conjecture for the four classical means. The inequality holds for every negative real number $n$ and every real number $n\geq\frac12$. Moreover, for every real numbe…

Quadratic-harmonic versus Seiffert mean conjecture. For all positive real numbers $x$ and $y$,

Let $a,b>0$ with $a eq b$, and let $Mt$ denote the power mean of order $t$ of $a$ and $b$. Let $P$ and $X$ denote the Seiffert and exponential means, respectively. **Best-constant…

Let $a,b>0$ with $a\neq b$. Denote by $A(a,b)$, $G(a,b)$, $L(a,b)$, and $\mathcal{I}(a,b)$ the arithmetic, geometric, logarithmic, and identric means, respectively, and by…

Toader–Qi/logarithmic-mean conjecture. The inequality

Logarithmic mean conjecture. If $x_1(t)=t$, $x_2(t)=t^2$, $\ldots$, $x_{n-1}(t)=t^{n-1}$, and $x_n(t)=\log t$, then

Vuorinen–Madras conjecture. For all $a,b>0$ with $a\ne b$,

Let $a,b>0$ with $a\ne b$. Denote by $N$ the Neuman–Sándor mean and by $T_p$ the second power-type Seiffert mean of $a$ and $b$. Sharp threshold conjecture. The inequality … holds…

Ideal monotonicity conjecture. The function $f_{i,p,s,t}$ is increasing with respect to $x$ if and only if $p\le -(i+1)$, and decreasing with respect to $x$ if and only if…

Let $M_{f,m_{1},m_{2}}$ be the means defined in the paper, with $m_{1}$ and $m_{2}$ denoting the interpolation orders, and let $f$ be a function on the relevant domain. Reciprocal-…

Let $a,b$ be positive real numbers, let $M_{p,m_{1},m_{2}}(a,b)$ denote the means defined in the paper, and set $n=m_{1}+m_{2}+1$. The harmonic mean is $H(a,b)=\tfrac{2ab}{a+b}$. A…