A question regarding pluriharmonic functions

Question

A real-valued function $u$ that is defined on a domain $D$ of $\mathbb{C}^n$ is pluriharmonic if

1. $u$ is of class $C^2$ and
2. for all $a\in D$ and $b\in\mathbb{C}^n$ the function $\lambda\mapsto u(a+\lambda b)$ is harmonic on the set $$D=\{\lambda\in\mathbb{C}: a+\lambda b\in D\}.$$

My question is: What if the second condition is satisfied only for $b\not=0$? Is $u$ still pluriharmonic? (the first condition is satisfied)

Answer 1

The condition is automatically satisfied for $b = 0$ because $\lambda \mapsto u(a)$ is a constant function and therefore harmonic.