On a sumset problem for integers

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$\mathrm{\mid }A+k\cdot A\mathrm{\mid }\ge (k+1)\mathrm{\mid }A\mathrm{\mid }-\lceil k(k+2)\mathrm{/}4\rceil$$ provided $\mathrm{\mid }A\mathrm{\mid }\ge (k-1{)}^{2}k!$, where $A+k\cdot A={a+kb:
a,b\in A}$\mathrm{.}Wealsoestablishtheinequality$|A+4\cdot A|\geq 5|A|-6 $for$|A|\geq 5$.