$L^p$ convergence of a sequence

Question

Given the sequence of functions $f_h (x)=x^{\frac {1}{h} } \log(x)$, for $ 0 <x<1$, for which $p\in [1,+ \infty [$ does it converge in $L^p $?

The pointwise limit of $f_h $ is the function $f(x)=\log (x) $. Can you help me to study $||x^{\frac {1}{h} } \log(x)- \log (x)||_p$?

Thanks to everybody

Answer 1

We have

1. for all $x\in (0,1)$, $ \lim_ {h\to+\infty}f_h\left(x\right)=f(x)$ and
2. for all $x\in (0,1)$ and all $h\gt 0$, $\left\lvert f_h(x)\right\rvert \leqslant \left\lvert f(x)\right\rvert$.