Let <$E F> be an entire sine-type function of exponential type <$E sigma~>>~0>, which is real-valued on <$E bold roman R> and has only real zeros <$E left { lambda sub k right }:~lambda sub k~<<~lambda sub k+1>, <$E k~<174>~bold roman Z>. We consider entire functions of exponential type <$E symbol Г~sigma>, which are real-valued on <$E bold roman R>, satisfying <$E f(x)~=~omicron (|x|)>, <$E x~symbol О~symbol С inf> and <$E (-1) sup k f( lambda sub k )~symbol У~0>, <$E k~<174>~bold roman Z>. It is proved that the zeros of such functions are real and simple except for points <$E left { lambda sub k right }>, each of which can be zeros of multiplicity not greater than 2. These results are applied to concrete classes of functions and to the problem of stability of entire functions. We also improve and complement some results due to Polya.

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