I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.



Entire function bounded at every line
Yes, there are such functions. Take a very narrow region $D$ containing the positive ray,
with nice boundary and such that $D$ intersects every any horizontal line other than the real line by a bounded interval.
Let $g$ be a conformal map of $D'$ onto the right half-plane, where
$D'\subset D$ is another similar region. Then with appropriate choice of $D$ and $D'$
$$f(z)=\int\_{\partial D'} \frac{e^{g(\zeta)}}{\zeta-z}d\zeta$$


will converge for $z$ outside $D$ and the function $f$ will be bounded outside $D$ and extend to an entire function (by a deformation of the contour).


For the details, see for example MR2753600 or MR0545054.


EDIT. The method is very flexible of course. Taking $D$ to be a half-strip and $g(z)=e^z$ one obtains a Mittag-Leffler function. Replacing it by $f(z+4i)$ you obtain a function that is bounded on every line from the origin. But to obtain a function as you ask, a half-strip $D$ will not work, so the function is less elementary.


With the same method one can also construct functions which tend to zero on every line: just replace $f$ by $f(z)/(z-z\_0)$ where $f(z\_0)=0$. Existence of infinitely many zeros of the original $f$ is easy to prove. Repeating this you can find a function which tends to $0$ on every line faster than any polynomial.


