Let \(f(z) = \displaystyle \sum \nolimits _{k = 1}^{\infty }f_k z^{k}\) be an entire transcendental function, let \((\lambda _n)\) be a sequence of positive numbers increasing to \( + \infty ,\) and let a series \(A(z) = \displaystyle \sum \nolimits _{n = 1}^{\infty }a_nf(\lambda _n z)\) be regularly convergent in \({\mathbb {D}} = \{z:|z|<1\}.\) We study the starlikeness and convexity of the function A. For example, if \(\displaystyle \sum \nolimits _{n = 1}^{\infty }\lambda ^{-\tau }_n = T< + \infty ,\) \(\ln |a_n|\le -e\lambda _n,\) and \(T\displaystyle \sum \nolimits _{k = 2}^{\infty }k|f_k| (k + \tau )^{k + \tau }\le \left| f_1\displaystyle \sum \nolimits _{n = 1}^{\infty }a_n\lambda _n\right| ,\) then the function A is starlike. It is proved that, under certain conditions imposed on the parameters, the differential equation \(z^2w'' + (\beta _0 z^2 + \beta _1z)w' + (\gamma _0z^2 + \gamma _1 z + \gamma _2) w = 0\) has an entire solution A, which is starlike or convex in \({\mathbb {D}}.\)
