Define the Nevanlinna class of holomorphic functions on unit disk by $\displaystyle N(\mathbb{D})=\left\{f\in\text{Hol}(\mathbb{D}):\sup_{0<r<1}\int_\mathbb{T}\log^+|f_r|dm<\infty\right\}$ where $\log^+=\max(0,\log t)$ for $t>0$ and $f_r(z)=f(rz)$ and $m$ is the normalized Lebesgue measure on unit circle $\mathbb{T}$ .
Let $f\in N(\mathbb{D})$ , $f(0)\neq0$ and $(\lambda_n)_{n\geq1}$ are the zeroes of $f$ in $\mathbb{D}$ counting multiplicity and a real singular measure $\mu$ w.r.t. $m$ satisfies $\displaystyle V_\mu(z)=\exp\left(-\int_\mathbb{T}\frac{\xi+z}{\xi-z}d\mu(\xi)\right)$ . Show that $$\log|f(0)|+\sum_{n\geq1}\frac{1}{\lambda_n}+\mu(\mathbb{T})=\int_\mathbb{T}\log|f|dm$$
A hint says to use the inner-outer factorization of Nevanlinna class like $f=\lambda BV_\mu[h]$ where $\lambda\in\mathbb{C}$ with $|\lambda|=1$ , $B$ is the Blaschke product of zeroes of $f$ and $\displaystyle[h](z)=\exp\left(\int_\mathbb{T}\frac{\xi+z}{\xi-z}\log|h(\xi)|dm(\xi)\right)$ and $h=|f|$ . Taking absolute and log both sides give $$\log|f/B|=\log|V_\mu|+\log|[h]|$$ $$\implies\log|f/B|=-\int_\mathbb{T}\frac{1-|z|^2}{|\xi-z|^2}d\mu(\xi)+\log|h|$$ $$\implies\log|(f/B)(0)|+\int_\mathbb{T}d\mu(\xi)=\log|h(0)|=\log|[f](0)|$$ $$\implies\log|f(0)|+\sum_{n\geq1}\log\frac{1}{\lambda_n}+\mu(\mathbb{T})=\int_\mathbb{T}\log|f|dm$$ a.e. on $\mathbb{T}$ . Everything is alright except the log is appearing inside summation part , and it is evident by mean value theorem , can someone throw some light what is possibly wrong here ? Thanks in advance .