Is there a non-constant entire function which is bounded on the real axis?

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Is there exist a nonconstant entire function which is bounded on the real axis ?

My attempt : I think yes take $f(z) = e^z$, now real axis have $y=0$ that $f(z) = e^{x}$

Is its true ?

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2 Answers

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How about the function $f(z)=\sin(z)$ ?

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With

$z = x + iy, \tag 1$

on the real axis $\Bbb R$ we have

$e^z = e^x, \tag 2$

which is clearly not bounded since

$\displaystyle \lim_{x \to \infty} e^x = \infty; \tag 3$

on the other hand, taking

$f(z) = e^{-z^2}, \tag 4$

on $\Bbb R$ we have

$f(z) = e^{-x^2} > 0, \; x \in \Bbb R, \tag 5$

attains its maximum value at $x = 0$, and also

$\displaystyle \lim_{x \to \pm \infty} e^{-x^2} = 0; \tag 6$

thus $f(z) = e^{-z^2}$ is manifestly bounded on the real axis, and it is clearly entire.

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