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Let $x,y,$ and $t$ be fixed real numbers, $0<x<y$, $0<t<1$. Does the following inequality hold for some $c$

$$\frac{\log{tx+(1-t)y}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{sw+(1-s)z}}{\log^s{w}\log^{(1-s)}{z}}$$

for $x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.

Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$

$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{\log^s{w}\log^{(1-s)}{z}}$$

for $1<x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.

Let $x,y,$ and $t$ be fixed real numbers, $0<x<y$, $0<t<1$. Does the following inequality hold

$$\frac{\log{tx+(1-t)y}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{sw+(1-s)z}}{\log^s{w}\log^{(1-s)}{z}}$$

for $x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.

Let $x,y,$ and $t$ be fixed real numbers, $0<x<y$, $0<t<1$. Does the following inequality hold for some $c$

$$\frac{\log{tx+(1-t)y}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{sw+(1-s)z}}{\log^s{w}\log^{(1-s)}{z}}$$

for $x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.

Let $x,y,$ and $t$ be fixed real numbers, $0<x<y$, $t>0$. Does the following inequality hold

$$\frac{\log{tx+(1-t)y}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{sw+(1-s)z}}{\log^s{w}\log^{(1-s)}{z}}$$

for $x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.

Let $x,y,$ and $t$ be fixed real numbers, $0<x<y$, $0<t<1$. Does the following inequality hold

$$\frac{\log{tx+(1-t)y}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{sw+(1-s)z}}{\log^s{w}\log^{(1-s)}{z}}$$

for $x<y\leq w<z$, $|z-w|\leq c|y-x|$, $c>0$, and $0<s<1$?

Intuition says this is true since $\log$ becomes ''flatter'' as its argument grows, so that the ratio on the right hand side should be closer to one when $z$ and $w$ are larger. Also how large can $c$ become?

Edit: I can more or less see how to do this with crude error bounds. I guess I am curious if there are simpler ways of seeing this.