We have the formula for pricing the call type of an option under the Black Scholes formula (European type) :
$$C(S_t,t;K,T)= S_t \cdot e^{-D(T-t)} \cdot\Phi(d_+) - K\cdot e^{-r(T-t)} \cdot\Phi(d_-)$$ where $d_{\pm}$ is given by :
$$d_{\pm} = \frac{\log(S_t/K)+(r-D\pm\frac{1}{2}\cdot \sigma^2) \cdot (T-t)}{\sigma \cdot \sqrt(T-t)} $$
The Vega measure of sensitivity is the first partial derivative of $C(S_t,t;K,T)$ with respect to $\sigma$ and is defined as : $$\mathcal{V}_{ega}= \sqrt{\frac{T-t}{2\pi}} S_t \cdot e^{-D \cdot (T-t)} \cdot e^{-\frac{1}{2} \cdot d_{+}^2}$$
I am trying to implement the Newton-Raphson method for the calculation of the implied (forward looking) volatility of an option. In order for the algorithm to ensure convergence we must find the inflation point, i.e the point that the function changes convexity or the interval that the function is convex and beyond that it becomes concave. For the Black Scholes formula this interval is (Manaster & Koehler Seed Value) : $$\left(0, \sqrt{ \frac{2 \cdot \log(\frac{S_t \cdot e^{r(T-t)}}{K} )}{T} } \right] $$. In order to find this interval one must implement the second derivative test to check in which point the $f''(x)=0$ (in general).For the Black Scholes this is the Vomma sensitivity measure which is the second derivative with respect to volatility $\sigma$ and is given by : $$\mathcal{V}_{omma}= \sqrt{\frac{T-t}{2\pi}} K \cdot e^{-r \cdot (T-t)} \cdot e^{-\frac{1}{2} \cdot d_{-}^2} \cdot \left(\frac{d_+ \cdot d_- }{2}\right)$$
My question is this interval was a result of such a calculation ? And if I am not correct how this interval is calculated ?