[LAD] Is this sort of window function used? (can't identify)

While experimenting with window functions for spectral analyzis, I compared Hann, Sin and Lanczos. It is easy to notice, that Hann is really same as sin(x)^2. Lanczos is tiny bit better, because its sides are tiny bit smoother, compared to sin(). It seems, that unsmoothed corners between sides and zero axis for sin() is reason why sides are so high, compared to Hann. Hamming and more over Gaussian have ideal smooth sides, but narrower middle (probably this one reason why central leaf is wider for them). Just for experiment i tried to change sin(x)^2 to just sin(x)^f, where 1.0f < f < 2.0f. And it looks like any f>1 causes derivative to be =0 at zero axis. The only thing, affected by exact amount in this range, is how fast it will become zero. While it is easy to notice with Hann example, factor around 1.2 or 1.1 make it hard to notice without very deep zoom. With f=1.25 or 1.26 it nearly reproduces Lanczos, thought difference may be noticed, if plotted at the same time. Though still not have enough precise integral for weakening correction, i noticed that side leafs falldown slightly faster than for Hann. Now I'm curious, is such function is in use? I don't know how to call it for search request. E.g., after reinventing Welch window by just multiplicating y=2x with y=2-2x, I already knew it is parabola. For sin(x)/x i know it is sinc. But what is sin(x)^y, at least at some 'y' between 1 and 2 ? I feel, that this is also something reinvented. Just like writing sin(x)^2, i discovered later that it is Hann. Need help. One of professors, who are still aware of signal processing stuff, adviced me to reed this book (found localized to russian): https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/ReferencesPap… but i still have to find time to learn it (besides of deepening my math knowledge).