Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we have $$(f(a+2))^\frac1k-(f(a))^\frac1k\geq 1$$ holds with $f(1)=1$?

Is there any tool to characterize such results?

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we have $$(f(a+2))^\frac1k-(f(a))^\frac1k=1$$ holds with $f(1)=1$?

Is there any tool to characterize such results?

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we have $$f(a+2)\geq ((f(a))^\frac1k+1)^k$$ holds with $f(1)=1$?

Is there any tool to characterize such results?

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we have $$(f(a+2))^\frac1k-(f(a))^\frac1k\geq 1$$ holds with $f(1)=1$?

Is there any tool to characterize such results?