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In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></semantics></math></inline-formula>) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.