On Hecke eigenvalues of cusp forms in almost all short intervals

Let $\psi$ be a function such that $\psi (x)\to \mathrm{\infty }$ as $x \rightarrow \infty.$Let$\lambda_{f}(n)$bethe$n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z})\mathrm{.}$ We show that for any real valued function $h(x)$ such that $(\log X{)}^{2-2\alpha }\ll h(X)=o(X),$ $$\sum _{n=x}^{x+h(X)}\mathrm{\mid }{\lambda }_f(n)\mathrm{\mid }{\ll }_{f}h(X)\psi (X)(\log X{)}^{\alpha -1}$$ for all but $O_f(X\psi (X{)}^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $\alpha$ is the average value of $\mathrm{\mid }{\lambda }_f(p)\mathrm{\mid }$ over primes. We generalize this for $\mathrm{\mid }{\lambda }_f(n){\mathrm{\mid }}^{2^k}$ for $k\in {\mathbb{Z}}^{+}\mathrm{.}$