| Summary: | Let <i>X</i> be a commutative normed algebra with a unit element <i>e</i> (or a normed field of characteristic different from 2), where the associated norm is sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo><mi>h</mi><mo>(</mo><mi>s</mi><mi>x</mi><mo>+</mo><mi>t</mi><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove the Hyers-Ulam stability of that functional equation under some additional conditions.
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