| Summary: | It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, by the tangential affine function that passes through the point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="double-struck">E</mi><mo>{</mo><mi>X</mi><mo>}</mo><mo>,</mo><mi>f</mi><mo>(</mo><mi mathvariant="double-struck">E</mi><mo>{</mo><mi>X</mi><mo>}</mo><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">E</mi><mo>{</mo><mi>X</mi><mo>}</mo></mrow></semantics></math></inline-formula> is the expectation of the random variable <i>X</i>. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to <i>f</i>, it turns out that when the function <i>f</i> is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="double-struck">E</mi><mo>{</mo><mi>X</mi><mo>}</mo><mo>,</mo><mi>f</mi><mo>(</mo><mi mathvariant="double-struck">E</mi><mo>{</mo><mi>X</mi><mo>}</mo><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>. In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as “Jensen-like” inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.
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