ORIGINAL RESEARCH PAPER On approximation properties of Gupta-type operators Neha Malik 1 Received: 2 September 2018 / Accepted: 22 August 2019 Ó Forum D’Analystes, Chennai 2019 Abstract Summation-integral type operators play an important role in the theory of approximation concerning linear positive operators. In this direction, Gupta pro- posed several examples of summation-integral type operators. In one of his recent papers, some examples of genuine operators have been proposed. We discuss cer- tain approximation properties of such operators and estimate direct results along with a Gru ¨ss-type Voronovskaya result. We obtain moments by deriving a recur- rence relation and establish the difference of the genuine operators with the Mas- troianni operators. Keywords Mastroianni operators  Gupta-type operators  Gru ¨ss-type Voronovskaya result  Difference of operators Mathematics Subject Classification 41A36  47A58 1 Introduction In the year 2011, for f 2 C½0; 1Þ; the generalized form of the Mastroianni operators was considered by Lo ´pez-Moreno et al. in [5], (also cf. [17]), defined in the following way: M n;c ðf ; xÞ¼ X 1 k¼0 ðxÞ k k! / ðkÞ n;c ðxÞ f k n  ; ð1Þ where / n;c ðxÞ has two special cases: & Neha Malik neha.malik_nm@yahoo.com 1 Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India 123 The Journal of Analysis https://doi.org/10.1007/s41478-019-00195-z