On the number of conjugacy classes of a finite solvable group

Let p be a prime that divides the order of the group G. We show that a finite solvable group has class number at least f(p) where \(f(p):=\min \{x+\frac{p-1}{x}: x\in \mathbb {N}, x \mid (p-1)\}\). We also obtain some applications to character degrees.