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In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo><</mo><mi>n</mi><mo>≤</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number <i>n</i> decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of <i>n</i>, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula–Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth <i>u</i> even though this inequality decreases (or increases) gradually as the depth <i>u</i> of the hyperbolic potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mn>1</mn></msub></semantics></math></inline-formula> (or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mn>2</mn></msub></semantics></math></inline-formula>) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth <i>u</i> of the potential wells increases, while the fractional derivative number <i>n</i> decreases.