Abstract: We consider a new class<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="script">C</mml:mi><mml:mi mathvariant="script">C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>of bounded doubly close-to-convex functions. Coefficient bounds, distortion theorems, and radius of convexity for the class<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi mathvariant="script">C</mml:mi><mml:mi mathvariant="script">C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>are investigated. A corresponding class of doubly close-to-starlike functions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msup><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="script">S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>is also considered.