Rational approximation with real, negative zeros and poles

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This note is concerned with the approximation of cosh √ x on [0, 1] by polynomials having only real negative zeros and by rational functions having only real negative zeros and poles. We establish here that cosh √ x can be approximated on [0, 1] by polynomials of degree n having only real negative zeros with an error ⩽ 4 n −1 but not better than c 1 n −1 ( c 1 some positive constant). Further, we show that cosh √ x cannot be approximated on [0, 1 ] by rational functions of total degree n having only real negative zeros and poles with an error better than c 2 n −4.5.