^{1}

^{2, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval

To approximate integrable functions on the interval

The general sequence of operators

Stancu [

The purpose of this paper is to introduce a new Stancu type generalization of the operators defined in [

In order to prove our main results, we need the following lemmas.

Let the

By definition of

Moments for

For sufficiently large

By using Cauchy-Schwarz inequality, it follows from Remark

Let

We give the proof for only first inequality, and the other is similar. Using Remark

Suppose

Using the identity

Throughout the paper by

the function

Let

Using the mean value theorem, we have

For estimating the integral

Combining (

Let

The results of our lemmas and theorems are more general rather than the results of any other previously proved lemmas and theorems, which will enrich the literature of applications of quantum calculus in operator theory and convergence estimates in the theory of approximations by positive linear operators. The researchers and professionals working or intend to work in areas of mathematical analysis and its applications will find this research paper to be quite useful. Consequently, the results so established may be found useful in several interesting situations appearing in the literature on mathematical analysis, pure and applied mathematics, and mathematical physics. Some interesting applications of the positive approximation linear operators can be seen in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research paper. Special thanks are due to Professor Józef Banaś, Editor of the Journal of Function Spaces, for his efforts to send the reports of the paper timely. The authors are also grateful to all the editorial board members and reviewers of esteemed journal, that is, Journal of Function Spaces. The second author Lakshmi Narayan Mishra acknowledges the Ministry of Human Resource Development (MHRD), New Delhi, India, for supporting this research paper at the Department of Mathematics, National Institute of Technology (NIT), Silchar, Assam. The third author Vishnu Narayan Mishra acknowledges that this paper project was supported by Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat (Gujarat), India. All the authors carried out the proof of theorems. Each author contributed equally in the development of the paper. Vishnu Narayan Mishra conceived of the study and participated in its design and coordination. All the authors read and approved the final version of paper.

^{r},

^{1}·

_{p}) operator of conjugate series of its Fourier series