On approximation of the rate of convergence of Fourier series in the generalized Hölder metric by Deferred Nörlund mean

Abstract

In this paper, we have studied an estimate of the rate of convergence of Fourier series in the generalized Hölder metric $H_{L_{r}}^{(\omega )}$ space by using deferred Nörlund mean and established some new results. Our results are more advanced that generalize and unify many other results available in the literature.

1 Introduction, definitions and motivation

Let $f(t)\in L_{r}[0,2\pi ]$ be $2\pi$-periodic. We have the Fourier series of f at $t=x$ be given by

$$\begin{aligned} \displaystyle \sum _{n=0}^{\infty } u_{n}(x)=\frac{1}{2}a_{0}+\displaystyle \sum _{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx) \end{aligned}$$

(1.1)

with its partial sum

$$\begin{aligned} S_{n}(f;x)=\frac{1}{\pi }\int _{-\pi }^{\pi }f(x+t)D_{n}(t)dt, \end{aligned}$$

where

$$\begin{aligned} D_{n}(t)=\frac{sin(n+\frac{1}{2})t}{2sin\frac{t}{2}}. \end{aligned}$$

By simple calculation (see [7, p. 50]),

$$\begin{aligned} S_{n}(f;x)-f(x)=\frac{2}{\pi }\int _{0}^{\pi }\phi _{x}(t)\frac{sin(n+\frac{1}{2})t}{2sin\frac{t}{2}}dt, \end{aligned}$$

(1.2)

where

$$\begin{aligned} \phi _{x}(t)=\frac{1}{2}\{f(x+t)+f(x-t)-2f(x)\}. \end{aligned}$$

Let $C_{2\pi }$ denote the Banach space of all $2\pi$-periodic continuous functions defined on $[0,2\pi ]$ under the supremum norm. The space $L_{r}[0, 2\pi ]$  $(r=\infty )$ reduces to $C_{2\pi }$ defined over $[0,2\pi ]$. We write

$$\begin{aligned} \Vert f\Vert _{c}= & {} \sup _{0 \le t \le 2\pi }|f(t)|~~(r=\infty );\\ \Vert f\Vert _{r}= & {} \left( \int _{0}^{2\pi }|f(t)|^{r}dt\right) ^{\frac{1}{r}}~~(r\ge 1);\\ \omega (\delta ;f)= & {} \sup _{0\le |h|\le \delta }\Vert f(x+h)-f(x)\Vert _{c}~~(f\in C_{2\pi });\\ \omega _{r}(\delta ;f)= & {} \sup _{0\le |h|\le \delta }\Vert f(x+h)-f(x)\Vert _{r},~~(f\in L_{r}[0,2\pi ])~(r\ge 1). \end{aligned}$$

The quantities $\omega (\delta ;f)$ and $\omega _{r}(\delta ;f)$ are called the modulus of continuity and integral modulus of continuity of f respectively. It is known ([7, p. 45]) that $\omega (\delta ;f)$ and $\omega _{r}(\delta ;f)$ both tend to zero as $\delta \rightarrow 0$.

For $(0<\alpha \le 1)$$(\delta >0)$ and let $H_{\alpha }$ be defined by

$$\begin{aligned} H_{\alpha }=\{f\in C_{2\pi }:\omega (\delta ;f)=O(\delta ^{\alpha })\}. \end{aligned}$$

(1.3)

It is known that $H_{\alpha }$ is a Banach space with norm,

$$\begin{aligned} \Vert f\Vert _{\alpha }=\Vert f\Vert _{c}+\sup (\delta ^{-\alpha }\omega (\delta ))~~(0<\alpha \le 1)~~(\delta >0) \end{aligned}$$

(1.4)

and

$$\begin{aligned} H_{\alpha }\subseteq H_{\gamma }\subseteq C_{_{2\pi }},~~(0\le \gamma <\alpha \le 1). \end{aligned}$$

The metric induced by the norm $\Vert .\Vert _{\alpha }$ on $\Vert H\Vert _{\alpha }$ is called the Hölder metric. Further, Leindler [5] generalized the Hölder metric $\Vert H\Vert _{\alpha }$ to obtain a space $\Vert H\Vert ^{\omega }$,

$$\begin{aligned} H^{\omega }=\{f\in C_{2\pi }:\omega (\delta ,f)=O(\omega (\delta ))\}, \end{aligned}$$

(1.5)

where $\omega$ is the modulus of continuity, that is, $\omega$ is a positive non-decreasing continuous function on $[0,2\pi ]$ with the properties:

• (i) $\omega (0)=0$;

• (ii) $\omega (\delta _{1}+\delta _{2})\le \omega (\delta _{1})+\omega (\delta _{2})$;

• (iii) $\omega (\lambda \delta )\le (\lambda +1)\omega (\delta )~~(\lambda \ge 0)$.

  Also, $H^{\omega }$ is a Banach space with the norm:

  $$\begin{aligned} \Vert f\Vert _{\omega }=\Vert f\Vert _{c}+\sup _{x,y}\frac{|f(x)-f(y)|}{\omega (|x-y|)}~~(x\ne y). \end{aligned}$$

  (1.6)

Here, as regards to Lipschitz classes we may recall that, a function $f\in Lip (\omega (t); r)$,

if

$$\begin{aligned} \Vert f(x+t)-f(x)\Vert _{r}=O(\omega (t))~~(r\ge 1),~t>0, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\Vert f(x+t)-f(x)\Vert _{r}}{\omega (t)}=A_{1}(f;\omega (t)). \end{aligned}$$

Consider the space (see [6]),

$$\begin{aligned} H_{r}^{(\omega )}=\{f\in L_{r}[0,2\pi ]:A_{1}(f;\omega (t))<\infty \}. \end{aligned}$$

(1.7)

It is also a Banach space with the norm:

$$\begin{aligned} \Vert f\Vert _{r}^{\omega }=\Vert f\Vert _{r}+A_{1}(f;\omega (t)). \end{aligned}$$

(1.8)

Furthermore, Das et al. [2] generalized a space,

$$\begin{aligned} H_{(\alpha ,r)}=\{f\in L_{r}:\Vert f(x+t)-f(x)\Vert _{r}=O(t^{\alpha })~~(r\ge 1),~t>0\}, \end{aligned}$$

(1.9)

under the norm

$$\begin{aligned} \Vert f\Vert _{(\alpha ,r)}=\Vert f\Vert _{r}+\frac{\Vert f(x+t)-f(x)\Vert _{r}}{(t^{\alpha })}, \end{aligned}$$

(1.10)

which is also a Banach space.

Similarly, $f\in W (L_{r},\omega (t))$,

if

$$\begin{aligned} \Vert (f(x+t)-f(x))\sin ^{\beta }x\Vert _{r}=O(\omega (t))~~(\beta >0) \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\Vert (f(x+t)-f(x))\sin ^{\beta }x\Vert _{r}}{(\omega (t))}=A_{2}(f;\omega (t)). \end{aligned}$$

Now, we define the space,

$$\begin{aligned} H_{L_{r}}^{(\omega )}=\{f\in L_{r}[0,2\pi ]:A_{2}(f;\omega (t))<\infty \}. \end{aligned}$$

(1.11)

It is also Banach space under the norm

$$\begin{aligned} \Vert f\Vert _{L_{r}}^{\omega }=\Vert f\Vert _{L_{r}}+A_{2}(f;\omega (t)). \end{aligned}$$

(1.12)

Next, in the spaces $H_{L_{r}}^{(\omega )}$ and $H_{L_{r}}^{(v)}$, if $\frac{\omega (t)}{v(t)}$ is non-decreasing, then

$$\begin{aligned} H_{L_{r}}^{(\omega )}\subseteq H_{L_{r}}^{(v)}\subseteq L_{r}~~(r\ge 1). \end{aligned}$$

(1.13)

Also,

$$\begin{aligned} \Vert f\Vert _{L_{r}}^{v}\le \max \left( 1,\frac{\omega (2\pi )}{v(2\pi )}\right) \Vert f\Vert _{L_{r}}^{(\omega )}. \end{aligned}$$

Next, for $(0\le \gamma <\alpha \le 1)$ if we put $\omega (t)=t^{\alpha }$, $v(t)=t^{\beta }$ and $\beta =0$ then (1.13) reduces to the following:

$$\begin{aligned} H_{(\alpha ,r)}\subseteq H_{(\gamma ,r)}\subseteq L_{r}~~(r\ge 1). \end{aligned}$$

(1.14)

Note that the space $H_{(\alpha ,\infty )}$ reduces to $H_{\alpha }$ space.

Remark 1

If $\omega (t)=t^{\alpha }~~(0<\alpha \le 1)$, then the space $H^{\omega }$ reduces to the $H_{\alpha }$ space (the norm $\Vert .\Vert _{\omega }$ being replaced by $\Vert .\Vert _{\alpha }$) and if $\omega (t)=t^{\alpha }~~(0<\alpha \le 1)$, then the space $H_{r}^{(\omega )}$ reduces to the $H_{(\alpha ,r)}$ space (with the norm $\Vert f\Vert _{r}^{(\omega )}$ replaced by $\Vert f\Vert _{(\alpha ,r)}$).

Remark 2

Let

$$\begin{aligned}H_{L_{r}}^{(\omega )}=\{f\in L_{r}[0,2\pi ]:A_{2}(f;\omega (t))<\infty \}.\end{aligned}$$

If $\beta =0$, then the space $H_{L_{r}}^{(\omega )}$ reduces to the $H_{r}^{(\omega )}$ space and if $\beta =0$, $\omega (t)=t^{\alpha }$ then the space $H_{L_{r}}^{(\omega )}$ reduces to the $H_{(\alpha ,r)}$ space.

Remark 3

If $\frac{\omega (t)}{t}\rightarrow 0$  $(t\rightarrow 0)$, then $f'(x)=0$ implies f is constant.

Let $(a_{n})$ and $(b_{n})$ be sequences of non-negative integers satisfying,

• (i) $a_{n}<b_{n}$  $(n\in {\mathbb {N}})$

  and

• (ii) $\displaystyle \lim _{n\rightarrow \infty }b_{n}=\infty$.

The deferred Cesàro mean, $D(a_{n},b_{n})$ is defined by Agnew [1, p. 414],

$$\begin{aligned} D_{n}(S_{n})=\frac{S_{a_{n}+1}+S_{a_{n}+2}+S_{a_{n}+3}+\cdots +S_{b_{n}}}{b_{n}-a_{n}}. \end{aligned}$$

(1.15)

In the notation of matrix transformation

$$\begin{aligned} D_{n}(S_{n})=\sum _{k=0}^{\infty }a_{n,k}S_{k}, \end{aligned}$$

where

$$\begin{aligned} a_{n,k}=\left\{ \begin{array}{ll} \frac{1}{q_{n}-p_{n}}&{} (p_{n}<k\le q_{n})\\ \\ 0 &{} (\text {otherwise}). \\ \end{array} \right. \end{aligned}$$

It is known that (Agnew [1]) $D(a_{n}, b_{n})$ is regular under the conditions (i) and (ii).

Let $(p_{n})$ be a sequence of positive real numbers. The deferred Nörlund mean $D_{a}^{b}N_{n}(f;x)$ is defined by Deǧer and Küçükaslan [4],

$$\begin{aligned} D_{a}^{b}N_{n}(f;x)=\frac{1}{P_{0}^{b_{n}-a_{n}-1}}\displaystyle \sum _{m=a_{n}+1}^{b_{n}}p_{b_{n}-m}S_{m}(f;x), \end{aligned}$$

(1.16)

where

$$\begin{aligned} P_{0}^{b_{n}-a_{n}-1}=\displaystyle \sum _{k=0}^{b_{n}-a_{n}-1}p_{k}\ne 0. \end{aligned}$$

Remark 4

If $b_{n} = n$ and $a_{n} =0$, then the $D_{a}^{b}N_{n}(f;x)$ mean in (2.16) reduces to the classical Nörlund mean. Moreover, if $p_{n} = 1$$(\forall ~ n)~(n\ge 0)$ in (1.16), then this yields (1.15), that is, the deferred Cesàro mean defined by Agnew [1]. Finally, if $b_{n}=n,~a_{n}=0$ in (1.15), then it reduces to the classical Cesàro mean of order 1.

We use the following notations:

$$\begin{aligned} \psi _{n}(x)= & {} D_{a}^{b}N_{n}(f;x)-f(x);\\ G_{x,y}(t)= & {} \phi _{x+y}(t)-\phi _{x}(t);\\ h= & {} h(P_{0}^{a_{n}})=\frac{\pi }{P_{0}^{a_{n}}}. \end{aligned}$$

2 Main result

In view of estimating the rate of convergence of Fourier series via the Deferred Nörlund Mean, we establish the following theorems based on generalized Hölder metric space $H_{L_{r}}^{\omega }$ in $W(L_{r},\omega (t))$-class.

Theorem 1

Let v and $\omega$ be moduli of continuity such that $(\frac{v}{\omega })$ is non-decreasing and $f\in H_{L_{r}}^{(\omega )}~~(r\ge 1)$.

Let

$$\begin{aligned} b_{n}=(2j+1)a_{n}+2j, \end{aligned}$$

(2.1)

where j is a positive integer. Then

$$\begin{aligned} {\text {(I)}}~~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}^{v}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})+O\left( \frac{p_{2j(a_{n}+1)+a_{n}-m}}{ P_{0}^{(j-1)+(a_{n})}}\right) \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}. \end{aligned}$$

If in addition, $\frac{\omega (t)}{tv(t)}$ is non-increasing, then

$$\begin{aligned} {\text {(II)}}~~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}^{v}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})\\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}} \end{aligned}$$

and a fortiori

$$\begin{aligned}&{\text {(III)}}~~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}^{v}=O(p_{2j(a_{n}+1)+a_{n}-m})+ O\left( \frac{\omega {\left( \frac{\pi }{P_{0}^{(j-1)+a_{n}}}\right) }}{v{\left( \frac{\pi }{P_{0}^{(j-1)+a_{n}}}\right) }}\right) . \end{aligned}$$

Theorem 2

Let $f\in W(L_{r},\omega (t))~~(r\ge 1)$ and let

$$\begin{aligned} b_{n}=(2j+1)a_{n}+2j, \end{aligned}$$

(2.2)

where j is a positive integer. Then

$$\begin{aligned} {\text {(I)}}~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})+O\left( \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\right) \nonumber \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{t^{3}}dt. \end{aligned}$$

If in addition, $\frac{\omega (t)}{t}$ is non-increasing, then

$$\begin{aligned} {\text {(II)}}~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})\nonumber \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{t^{3}}dt \end{aligned}$$

and a fortiori

$$\begin{aligned}&{\text {(III)}}~\Vert D_{a}^{b}(N_{n}(f;x))-f(x)\Vert _{L_{r}}=O(p_{2j(a_{n}+1)+a_{n}-m})+ O\left( \omega {\left( \frac{\pi }{P_{0}^{(j-1)+a_{n}}}\right) }\right) . \end{aligned}$$

We need the following lemma.

Lemma 1

Let v and $\omega$ be moduli of continuity such that $(\frac{v}{\omega })$ is non-decreasing. Then for $0\le t\le \pi$ and $r\ge 1$,

• (i) $\text {(i)}~~\Vert \phi _{x}(t)\Vert _{L_{r}}\le 2\omega (t);$

• (ii) $\Vert \phi _{x}(t)-\phi _{x+y}(t)\Vert _{L_{r}}=O(1)\{\omega (t), \omega (|y|)\};$

• (iii) $\Vert \phi _{x}(t)-\phi _{x+y}(t)\Vert _{L_{r}}=O(1) \omega (|y|)\frac{\omega (t)}{v(t)};$

• (iv) $\Vert \phi _{x}(t)-\phi _{x+y}(t)-\phi _{x}(t+h)+\phi _{x+y}(t+h)\Vert _{L_{r}}=O(1) \omega (|y|)\frac{\omega (h)}{v(h)}~~\left( h=\frac{\pi }{P_{0}^{(j-1)+a_{n}}}\right) .$

Proof

It can be proved in the similar line as in (see [3]). $\square$

Proof of the Theorem 1

Let $S_{n}(x)$ be the partial sum of Fourier series at x and let $D_{a}^{b}N_{n}(f;x)$ be the deferred Nörlund Mean of Fourier series of the form (1.16), then it can be easily verified that

$$\begin{aligned} D_{a}^{b}N_{n}(f;x)=\frac{1}{2\pi }\int _{0}^{\pi } \frac{p_{b_{n}-m}\sin P_{0}^{\frac{(b_{n}+a_{n}+2)}{2}}t\sin P_{0}^{\frac{(b_{n}-a_{n}-1)}{2}}t}{P_{0}^{b_{n}-a_{n}-1}\sin ^{2}\frac{t}{2}} \{f(x+t)+f(x-t)\}dt \end{aligned}$$

(2.3)

which reduces to (by virtute of condition (2.1)),

$$\begin{aligned} D_{a}^{b}N_{n}(f;x)=\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{0}^{\pi } \frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}} \{f(x+t)+f(x-t)\}dt. \end{aligned}$$

(2.4)

It may be easily verified that

$$\begin{aligned} \frac{2}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi }\phi _{x}(t)\frac{\sin P_{0}^{ja_{n}}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}dt=1, \end{aligned}$$

(2.5)

and hence

$$\begin{aligned} \psi _{n}(x)=D_{a}^{b}(N_{n}(f;x))-f(x)=\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{0}^{\pi } \phi _{x}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(2\sin \frac{t}{2})^{2}}dt. \end{aligned}$$

(2.6)

Clearly, as per the considered metric in the definition, we have

$$\begin{aligned} \Vert \psi \Vert _{L_{r}}^{(\omega )}=\Vert \psi \Vert _{L_{r}}+\sup _{y\ne 0} \frac{\Vert (\psi _{n}(x)-\psi _{n}(x+y))\sin ^{\beta }x\Vert _{L_{r}}}{\omega (|y|)}. \end{aligned}$$

(2.7)

Now,

$$\begin{aligned}&(\psi _{n}(x+y)-\psi _{n}(x))\sin ^{\beta }x\nonumber \\&\quad =\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi } \{\phi _{x+y}(t)-\phi _{x}(t)\}\frac{\sin ^{\beta }x\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(2\sin \frac{t}{2})^{2}}dt\nonumber \\&\quad =\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi } \{G_{x,y}(t)\}\frac{\sin ^{\beta }x\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}dt\nonumber \\&\quad =\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{h} \{G_{x,y}(t)\}\frac{\sin ^{\beta }x\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}dt\nonumber \\&\qquad +\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi } \{G_{x,y}(t)\}\frac{\sin ^{\beta }x\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}dt\nonumber \\&\quad =I+J~~(\text {say}). \end{aligned}$$

(2.8)

Also, by simple calculation

$$\begin{aligned} \frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}=O(P_{0}^{(j-1)+a_{n}})^{2}~~(0<t<\pi ). \end{aligned}$$

Now (for I), by the generalized Minkowski inequality and for $(a_{n}\ge 1)$ we have

$$\begin{aligned} \Vert I\Vert _{L_{r}}\le & {} \frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{h} \sin ^{\beta }x\Vert G_{x,y}(t)\Vert _{p}\left| \frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(2\sin \frac{t}{2})^{2}}\right| dt\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{ P_{0}^{(j-1)+(a_{n})}}(v|y|)(P_{0}^{(j-1)+(a_{n})})^{2}\int _{0}^{h}\sin ^{\beta }t\frac{\omega (t)}{v(t)}dt~(\text {by (iii) of Lemma}~1\mathrm{)}\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m}P_{0}^{(j-1)+(a_{n})}(v|y|)h\frac{\omega (h)}{v(h)}\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m}(v|y|)\frac{\omega (h)}{v(h)}. \end{aligned}$$

(2.9)

Next,

$$\begin{aligned} J= & {} \frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi }\sin ^{\beta }t G_{x,y}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(2\sin \frac{t}{2})^{2}}dt\\= & {} \frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi } \sin ^{\beta }t G_{x,y}(t)\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t\left\{ \frac{1}{(2\sin \frac{t}{2})^{2}}-\frac{1}{t^{2}}\right\} dt\\&+\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi } \sin ^{\beta }tG_{x,y}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{t^{2}}dt\\= & {} J_{1}+J_{2} ~~(\text {say}). \end{aligned}$$

Furthermore, since $\left\{ \frac{1}{(2\sin \frac{t}{2})^{2}}-\frac{1}{t^{2}}\right\}$ is bounded, we get

$$\begin{aligned} \Vert J_{1}\Vert _{L_{r}}\le & {} \frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi }\sin ^{\beta }t \Vert G_{x,y}(t)\Vert _{p}dt\nonumber \\= & {} O(1)\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}(v|y|)\int _{h}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}dt~(\text {by Lemma}~1\mathrm{)}\nonumber \\= & {} O(1)\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}(v|y|). \end{aligned}$$

(2.10)

Next,

$$\begin{aligned} J_{2}=\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi } \sin ^{\beta }t G_{x,y}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt. \end{aligned}$$

(2.11)

Replacing t by $t+h$ in (2.11), for any positive integer j, we obtain

$$\begin{aligned} J_{2}=-\frac{2p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi -h}\sin ^{\beta }t G_{x,y}(t+h)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(t+h)^{2}}dt. \end{aligned}$$

(2.12)

Clearly, from (2.11) and (2.12), it follows that

$$\begin{aligned} J_{2}= & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi } \sin ^{\beta }tG_{x,y}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{t^{2}}dt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi -h}\sin ^{\beta }t G_{x,y}(t+h)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(t+h)^{2}}dt\nonumber \\= & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi } \sin ^{\beta }tG_{x,y}(t)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{h} \sin ^{\beta }tG_{x,y}(t+h)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(t+h)^{2}}dt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{\pi }\sin ^{\beta }t G_{x,y}(t+h)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(t+h)^{2}}dt\nonumber \\&+\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{\pi -h}^{\pi } \sin ^{\beta }tG_{x,y}(t+h)\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(t+h)^{2}}dt\nonumber \\= & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{h}^{\pi }\left\{ \frac{G_{x,y}(t)}{t^{2}}-\frac{G_{x,y}(t+h)}{(t+h)^{2}}\right\} \sin ^{\beta }t\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}tdt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{0}^{h} G_{x,y}(t+h)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+a_{n}}t}{(t+h)^{2}}dt\nonumber \\&+\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+a_{n}}}\int _{\pi -h}^{\pi } G_{x,y}(t+h)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(t+h)^{2}}dt\nonumber \\= & {} K-L+M~~(\text {say}). \end{aligned}$$

Moreover, since

$$\begin{aligned} \sin P_{0}^{j(a_{n})}t\sin P_{0}^{j-1(a_{n})}t=O((P_{0}^{(j-1)+a_{n}})^{2}t^{2})~~(0<t<\pi ); \end{aligned}$$

so, it follows that

$$\begin{aligned} \Vert L\Vert _{L_{r}}\le & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{0}^{h}\Vert G_{x,y}(t+h)\Vert _{p}\sin ^{\beta }t\frac{(P_{0}^{(j-1)+a_{n}})^{2}t^{2}}{(t+h)^{2}}dt\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{ P_{0}^{(j-1)+(a_{n})}}(v|y|)\int _{0}^{h}\sin ^{\beta }t\frac{\omega (t+h)}{v(t+h)}(P_{0}^{(j-1)+(a_{n})})^{2}dt\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m} P_{0}^{(j-1)+(a_{n})}(v|y|)\int _{0}^{h}\sin ^{\beta }t\frac{\omega (t+h)}{v(t+h)}dt\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m} P_{0}^{(j-1)+(a_{n})}(v|y|)\frac{\omega (2h)}{v(2h)}\int _{0}^{h}\sin ^{\beta }tdt\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m} P_{0}^{(j-1)+(a_{n})}(v|y|)\frac{\omega (h)}{v(h)}h\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m} (v|y|)\frac{\omega (h)}{v(h)} ~ \text {as} ~(\omega (2h)\le 3\omega (h))~\text {and} ~(v(2h)\ge v(h).)\nonumber \\ \end{aligned}$$

(2.13)

Again, since

$$\begin{aligned} |\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t|\le 1~~(\pi -h<t<\pi ); \end{aligned}$$

so, it follows that

$$\begin{aligned} \Vert M\Vert _{L_{r}}\le & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{\pi -h}^{\pi }\Vert G_{x,y}(t+h)\Vert _{p}\sin ^{\beta }t\frac{dt}{(t+h)^{2}}\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}(v|y|)\int _{\pi -h}^{\pi }\sin ^{\beta }t\frac{\omega (t+h)}{v(t+h)}\frac{dt}{(t+h)^{2}}\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}(v|y|)\int _{\pi }^{\pi +h}\sin ^{\beta }t\frac{\omega (\theta )}{v(\theta )}\frac{d\theta }{(\theta )^{2}}\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}(v|y|). \end{aligned}$$

(2.14)

Next,

$$\begin{aligned} K= & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }G_{x,y}(t)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }G_{x,y}(t+h)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt\nonumber \\&+\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)(a_{n})}}\int _{h}^{\pi }G_{x,y}(t+h)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt\nonumber \\&-\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }G_{x,y}(t+h)\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{(t+h)^{2}}dt\nonumber \\= & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }\{G_{x,y}(t)-G_{x,y}(t+h)\}\sin ^{\beta }t\frac{\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t}{t^{2}}dt\nonumber \\&+\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }G_{x,y}(t+h)\sin ^{\beta }t\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t\left\{ \frac{1}{t^{2}}-\frac{1}{(t+h)^{2}}\right\} dt\nonumber \\= & {} K_{1}+K_{2}~~(say). \end{aligned}$$

Also, since

$$\begin{aligned} |\sin P_{0}^{j(a_{n})}t\sin P_{0}^{(j-1)+(a_{n})}t|\le 1 ~~(h<t<\pi ); \end{aligned}$$

so, using condition (iv) of Lemma 1, we get

$$\begin{aligned} \Vert K_{1}\Vert _{L_{r}}\le & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }\Vert G_{x,y}-G_{x+y}\Vert _{p}\sin ^{\beta }t\frac{dt}{t^{2}}\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}(v|y|)\frac{\omega (h)}{v(h)}\int _{h}^{\pi }\sin ^{\beta }t\frac{dt}{t^{2}}\nonumber \\= & {} O(1)p_{2j(a_{n}+1)+a_{n}-m}(v|y|)\frac{\omega (h)}{v(h)}. \end{aligned}$$

(2.15)

Next,

$$\begin{aligned} \Vert K_{2}\Vert _{L_{r}}\le & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{ P_{0}^{(j-1)+(a_{n})}}\int _{h}^{\pi }\Vert G_{x+y}(t+h)\Vert \sin ^{\beta }t\left\{ \frac{1}{t^{2}}-\frac{1}{(t+h)^{2}}\right\} dt\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{ (P_{0}^{(j-1)+(a_{n})})^{2}}(v|y|)\int _{h}^{\pi }\sin ^{\beta }t\frac{\omega (t+h)}{v(t+h)}\frac{(t+h)^{2}-t^{2}}{t^{2}(t+h)^{2}}dt\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}}(v|y|)\int _{h}^{\pi } \sin ^{\beta }t\frac{\omega (t+h)}{v(t+h)}\frac{dt}{t^{3}}\nonumber \\= & {} O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}}(v|y|) \int _{h}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}. \end{aligned}$$

(2.16)

Collecting the results from (2.9)–(2.16), we obtain

$$\begin{aligned} \sup _{y\ne 0}\frac{\Vert \psi _{n}(x)-\psi _{n}(x+y)\Vert _{L_{r}}}{(v|y|)}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})+O\left( \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\right) \nonumber \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}.\nonumber \\ \end{aligned}$$

(2.17)

Similarly, it can be proved that

$$\begin{aligned} \Vert \psi _{n}(x)\Vert _{L_{r}}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})+O\left( \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\right) \nonumber \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}. \end{aligned}$$

(2.18)

Hence, collecting the results of (2.7), (2.17) and (2.18), we obtain

$$\begin{aligned} \Vert \psi _{n}(x)\Vert _{r}^{v}= & {} O(p_{2j(a_{n}+1)+a_{n}-m})+O\left( \frac{p_{2j(a_{n}+1)+a_{n}-m}}{\pi P_{0}^{(j-1)+(a_{n})}}\right) \nonumber \\&+O(1)\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}, \end{aligned}$$

(2.19)

which establishes the proof (I) of Theorem 1.

If in addition $\frac{\omega (t)}{tv(t)}$ is non-increasing, then we have

$$\begin{aligned} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}\ge & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}}\frac{\omega (\pi )}{\pi v(\pi )} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{dt}{t^{2}}\nonumber \\\ge & {} \frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}}. \end{aligned}$$

(2.20)

Furthermore,

$$\begin{aligned}&\frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}} \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}\nonumber \\&\quad \le \frac{p_{2j(a_{n}+1)+a_{n}-m}}{(P_{0}^{(j-1)+(a_{n})})^{2}}\frac{(P_{0}^{(j-1)+(a_{n})})}{\pi } \frac{\omega \left( \frac{\pi }{P_{0}^{(j-1)+(a_{n})}}\right) }{ v\left( \frac{\pi }{P_{0}^{(j-1)+(a_{n})}}\right) } \int _{\frac{\pi }{P_{0}^{(j-1)+a_{n}}}}^{\pi }\sin ^{\beta }t\frac{dt}{t^{2}}\nonumber \\&\quad =O\left( \frac{\omega \left( \frac{\pi }{P_{0}^{(j-1)+(a_{n})}}\right) }{ v\left( \frac{\pi }{P_{0}^{(j-1)+(a_{n})}}\right) }\right) . \end{aligned}$$

(2.21)

Using (2.20) and (2.21) in (2.19), we obtain (II) and (III) of Theorem 1.

This completes the proof of Theorem 1. $\square$

Proof of the Theorem 2

Proceeding in the similar line as in Theorem 1 and using the fact,

$$\begin{aligned} \Vert \phi _{x}(t)\Vert _{L_{r}}=\Vert f(x+t)-f(x)\sin ^{\beta } x\Vert _{L_{r}}=O(\omega (t))~\text {~for}~~f\in W(L_{r},\omega (t))~~(r\ge 1) \end{aligned}$$

we can prove Theorem 2. $\square$

3 Corollaries and observations

Directly, as the consequences of Theorems 1 and 2, we have the following Corollaries.

Corollary 1

Let v and $\omega$ be moduli of continuity such that $(\frac{v}{\omega })$ is non-decreasing and $f\in H_{L_{r}}^{(\omega )}~~(r\ge 1)$ and let

$$\begin{aligned} b_{n}=(2j+1)a_{n}+2j, \end{aligned}$$

(3.1)

where j is a positive integer. Then

$$\begin{aligned} {\text {(I)}}~\Vert D_{0}^{n}(N_{n}(f;x))-f(x)\Vert _{r}^{v}= & {} O(p_{2j-m})+O\left( \frac{p_{2j-m}}{ P_{0}^{(j-1)}}\right) \\&+O(1)\frac{p_{2j-m}}{(P_{0}^{(j-1)})^{2}} \int _{\frac{\pi }{P_{0}^{j-1}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}}. \end{aligned}$$

If in addition $\frac{\omega (t)}{tv(t)}$ is non-increasing then

$$\begin{aligned}&{\text {(II)}}~\Vert D_{0}^{b}(N_{n}(f;x))-f(x)\Vert _{r}^{v}=O(p_{2j-m}) +O(1)\frac{p_{2j-m}}{(P_{0}^{(j-1)})^{2}} \int _{\frac{\pi }{P_{0}^{j-1}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)}{v(t)}\frac{dt}{t^{3}} \end{aligned}$$

and a fortiori

$$\begin{aligned}&{\text {(III)}}~\Vert D_{0}^{b}(N_{n}(f;x))-f(x)\Vert _{r}^{v}=O(p_{2j-m})+ O\left( \frac{\omega {\left( \frac{\pi }{P_{0}^{j-1}}\right) }}{v{\left( \frac{\pi }{P_{0}^{j-1}}\right) }}\right) . \end{aligned}$$

Proof

By substituting $b_{n}=n,~a_{n}=0$ in the associated conditions and by using Theorem 1, the proof of Corollary 1 follows. $\square$

Corollary 2

Let v and $\omega$ be moduli of continuity such that $(\frac{v}{\omega })$ is non-decreasing and $f\in W(L_{r},\omega (t)), ~r\ge 1$ and let

$$\begin{aligned} b_{n}=(2j+1)a_{n}+2j, \end{aligned}$$

(3.2)

where j is a positive integer. Then

$$\begin{aligned} {\text {(I)}}~\Vert D_{0}^{n}(N_{n}(f;x))-f(x)\Vert _{r}^{v}= & {} O(p_{2j-m})+O\left( \frac{p_{2j-m}}{ P_{0}^{(j-1)}}\right) \\&+O(1)\frac{p_{2j-m}}{(P_{0}^{(j-1)})^{2}} \int _{\frac{\pi }{P_{0}^{j-1}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)dt}{t^{3}}. \end{aligned}$$

If in addition $\frac{\omega (t)}{tv(t)}$ is non-increasing then

$$\begin{aligned}&{\text {(II)}}~\Vert D_{0}^{b}(N_{n}(f;x))-f(x)\Vert _{r}^{v}=O(p_{2j-m}) +O(1)\frac{p_{2j-m}}{(P_{0}^{(j-1)})^{2}} \int _{\frac{\pi }{P_{0}^{j-1}}}^{\pi }\sin ^{\beta }t\frac{\omega (t)dt}{t^{3}} \end{aligned}$$

and a fortiori

$$\begin{aligned}&{\text {(III)}}~\Vert D_{0}^{b}(N_{n}(f;x))-f(x)\Vert _{r}^{v}=O(p_{2j-m})+ O\left( \omega {\left( \frac{\pi }{P_{0}^{j-1}}\right) }\right) . \end{aligned}$$

Proof

By substituting $b_{n}=n,~a_{n}=0$ in the associated conditions and by using Theorem 2, the proof of Corollary 2 follows. $\square$

In the concluding remarks we can say that, the results established here generalizes the earlier results of Nayak et al. [6], Das et al. [3] and Leindler [5]. Further, our estimation via deferred weighted mean is a sharper estimation than those obtained from the sequence of partial sums of Fourier series.