I have a question that consists of the characterization of all functions $f(x)$ and all constants $k\in\mathbb{R}$ satisfying:
$$f:\mathbb{R}^+\rightarrow (0,1)$$
$$k-\int_4^x\frac{f(t)}{t}dt\leq\log(2)-\frac{1}{2}\log(x),\ \ \forall x\geq 4$$
Does someone have an idea about the second inequatily? Thanks a lot for your help!
I have found a sufficient condition. If we express $\log(x)$ by integral, then we have
$$k-\int_4^x\frac{f(t)}{t}dt\leq-\int_4^x\frac{1}{2t},\ \ \forall x\geq 4$$
which leads to a sufficient condition
$$k\leq 0$$
$$f(x)\geq\frac{1}{2}$$
Does someone have another idea to make this characterizaton more accurate? Thanks a lot