Is there an equivalent BS volatility formula for the Heston model, something like Hagan's formula for the SABR model? Of course, such a formula will be an approximation as in Hagan's formula.
Under the Heston model, we can price European options with the inverse Fourier transform (or FFT) quite precisely. So it's possible to numerically invert the price to the BS volatility. Nevertheless, an analytic volatility formula (albeit approximation) will be still helpful for many occasions. For example, FFT method seems unstable for extreme inputs (deep out-of-the-money or short time-to-maturity). See this question.