Let ${ ho}(A)$ and ${ ho}(B)$ denote the orders of entire functions $A(z)$ and $B(z)$ respectively. Suppose that ${ ho}(A)$ > 1 and 0 < ${ ho}(B){leq}frac{1}{2}$, and that ${ ho}$(A) is not an integer. Then it is shown that every nonconstant solution $f$ of $f^{{prime}{prime}}+A(z)f^{prime}+B(z)f=0$ is of infinite order if all the zeros of $A(z)$ lie in a certain angular sector depending on its genus. In addition, we investigate some growth properties of $A(z)$.