Page 69 of $\textbf{[1]}$, one finds the theorem 3.8 (Tumura-Hayma):

Suppose that $~f(z)~$ is meromorphic and has only a finite number of poles in the plane, and that $~f(z)~$ and $~f^{(l)}(z)~$ have only a finite number of zeros for some integer $~l\geq 2.~$ Then : $$f(z)=\dfrac{P_1(z)}{P_2(z)}e^{P_3(z)}$$ where $~P_1,~P_2,~P_3~$ are polynomials. If, further, $~f(z)~$ and $~f^{(l)}(z)~$ have no zeros, then $$f(z)=e^{Az+B}~~~~or~~~~f(z)=\dfrac{1}{(Az+B)^n}$$ where $~A,~B~$ are constants such that $~A\neq 0~$ and $~n~$ is a positive integer.}

$\textbf{QUESTION}$ : May we hope for a ``simple" proof of the following particular case of 3.8 (which is also theorem 5, page 22 of $\textbf{[2]})$ ?

Suppose that $~f(z)~$ is an entire function, and that $~f(z)~$ and $~f^{(2)}(z)~$ have no zeros in the plane. Then $~f(z)=e^{Az+B}~$ where $~A,~B~$ are constants such that $~A\neq 0.~$

$\textbf{[1]}$ ``Meromorphic Fuctions", W.K. Hayman, 1964, Oxford Mathematical Monographs

$\textbf{[2]}$ ``Picard values of Meromorphic Functions and their Derivatives", W.K. Hayman, Annals of Mathematics, Second Series, Vol. 70, No 1 (Jul.,1959), pp. 9-42