Show $|f(0)| \le e$ for holomorphic function satisfying $ |f(e^{i\pi t})|\leq e^{t}$

The function $$ g(z) = f(z) \cdot \overline{f(\bar z)} $$ is holomorphic in the closed unit disk and satisfies $$ |g(e^{i\pi t})| = |f(e^{i\pi t})| \cdot |f(e^{i\pi (2-t)})| \le e^t \cdot e^{2-t} = e^2 \, . $$ for $0 \le t \le 2$. The maximum modulus principle then gives $$ |f(0)|^2 = |g(0)| \le e^2 \, . $$

Alternatively one can use that $\log|f(z)|$ is subharmonic and therefore satisfies the mean-value inequality $$ \log |f(0)| \le \frac 12 \int_0^2 \log |f(e^{i\pi t})| \, dt \le \frac 12 \int_0^2 t \, dt = 1 \, . $$

The same estimate also follows from Jensen's formula.