Open Access

Approximation by a complex q-Baskakov-Stancu operator in compact disks

Journal of Inequalities and Applications20142014:249

https://doi.org/10.1186/1029-242X-2014-249

Received: 25 January 2014

Accepted: 24 May 2014

Published: 18 July 2014

Abstract

In this paper, we consider a complex q-Baskakov-Stancu operator and study some approximation properties. We give a quantitative estimate of the convergence, Voronovskaja-type result and exact order of approximation in compact disks.

MSC:30E10, 41A25, 41A28.

Keywords

complex q-Baskakov-Stancu operatordivided differencescomplex approximation

1 Introduction

Recently complex approximation operators have been studied intensively. For this approach, we refer to the book of Gal [1], where he considers approximation properties of several complex operators such as Bernstein, q-Bernstein, Favard-Szasz-Mirakjan, Baskakov and some others. Also we refer to the useful book of Aral, Gupta and Agarwal [2] who consider many applications of q-calculus in approximation theory. Now, for the construction of the new operators, we give some notations on q-analysis [3, 4].

Let q > 0 . The q-integer [ n ] and the q-factorial [ n ] ! are defined by
[ n ] : = [ n ] q = { 1 q n 1 q , q 1 , n , q = 1 for  n N
and
[ n ] ! : = { [ 1 ] q [ 2 ] q [ n ] q , n = 1 , 2 , , 1 , n = 0 for  n N  and  [ 0 ] ! = 1 ,
respectively. For integers n r 0 , the q-binomial coefficient is defined as
[ n r ] q = [ n ] q ! [ r ] q ! [ n r ] q ! .
The q-derivative of f ( z ) is denoted by D q f ( z ) and defined as
D q f ( z ) : = f ( q z ) f ( z ) ( q 1 ) z , z 0 , D q f ( 0 ) = f ( 0 ) ,
also
D q 0 f : = f , D q n f : = D q ( D q n 1 f ) , n = 1 , 2 ,
q-Pochhammer formula is given by
( x , q ) 0 = 1 , ( x , q ) n = k = 0 n 1 ( 1 q k x )
with x R , n N { } . The q-derivative of the product and the quotient of two functions f and g are
D q ( f ( z ) g ( z ) ) = f ( z ) D q ( g ( z ) ) + g ( q z ) D q ( f ( z ) )
and
D q ( f ( z ) g ( z ) ) = g ( z ) D q ( f ( z ) ) f ( z ) D q ( g ( z ) ) g ( z ) g ( q z ) ,
respectively (see in [3]). Moreover, we have
[ x 0 , x 1 , , x m ; f g ] = i = 0 m [ x 0 , x 1 , , x i ; f ] [ x i , x i + 1 , , x m ; g ] ,
(1.1)

where [ x 0 , x 1 , , x m ; f ] denotes the divided difference of the function f on the knots x 0 , x 1 , , x m (see [4] also [5]).

In [6], Aral and Gupta constructed the q-Baskakov operator as
Z n q ( f ) ( x ) = k = 0 [ n + k 1 k ] q k ( k 1 ) 2 z k ( x , q ) n + k 1 f ( [ k ] q k 1 [ n ] ) , n N ,
where x 0 , q > 0 and f is a real-valued continuous function on [ 0 , ) . The authors studied the rate of convergence in a polynomial weighted norm and gave a theorem related to monotonic convergence of the sequence of operators with respect to n. Not only they proved a kind of monotonicity by means of q-derivative but also they expressed the operator in terms of divided differences as follows:
W n , q ( f ) ( x ) = j = 0 [ n + j 1 ] ! [ n 1 ] ! q j ( j 1 ) 2 [ 0 , 1 [ n ] , [ 2 ] q [ n ] , , [ j ] q j 1 [ n ] ; f ] x j [ n ] j
(1.2)
n N , similar to the case of classical Baskakov operators in the sense of Lupaş in [7]. That is to say, Z n q ( f ) ( x ) = W n , q ( f ) ( x ) for x 0 and q > 0 , so they proved that
[ 0 , 1 [ n ] , [ 2 ] q [ n ] , , [ j ] q j 1 [ n ] ; f ] = q j ( j 1 ) q j f ( 0 ) [ j ] ! [ n ] j = f ( j ) ( ζ ) j ! , ζ ( 0 , [ j ] q j 1 [ n ] ) ,
(1.3)
where q r stands for q-divided differences given by q 0 f ( x j ) ,
q r + 1 f ( x j ) = q r q r f ( x j + 1 ) q r f ( x j )

for r N { 0 } .

A different type of the q-Baskakov operator was also given by Aral and Gupta in [8]. In [9] Finta and Gupta studied the q-Baskakov operator Z n q ( f ) ( x ) for 0 < q < 1 . Using the second-order Ditzian-Totik modulus of smoothness, they gave direct estimates. They also introduced the limit q-Baskakov operator.

In [10] Gupta and Radu introduced a q-analogue of Baskakov-Kantorovich operators and studied weighted statistical approximation properties of them for 0 < q < 1 . They also obtained some direct estimations for error with the help of weighted modulus of smoothness. Moreover, Durrmeyer-type modifications of q-Baskakov operators were studied in [11] and [12]. In [13], Söylemez, Tunca and Aral defined a complex form of q-Baskakov operators by
W n , q ( f ) ( z ) = j = 0 [ n + j 1 ] ! [ n 1 ] ! q j ( j 1 ) 2 [ 0 , 1 [ n ] , [ 2 ] q [ n ] , , [ j ] q j 1 [ n ] ; f ] z j [ n ] j
(1.4)

for q > 1 , f : D ¯ R [ R , ) C , replacing x by z in the operator W n , q ( f ) ( x ) given by (1.2). They obtained a quantitative estimate for simultaneous approximation, Voronovskaja-type result and degree of simultaneous approximation in compact disks.

In recent years, a Stancu-type generalization of the operators has been studied. Büyükyazıcı and Atakut considered a Stancu-type generalization of the real Baskakov operators in [14]. Also in [15], q-Baskakov-Beta-Stancu operators were introduced. In [16] Gupta-Verma studied the Stancu-type generalization of complex Favard-Szasz-Mirakjan operators and established some approximation results in the complex domain. In [17] Gal, Gupta, Verma and Agrawal introduced complex Baskakov-Stancu operators and studied Voronovskaja-type results with quantitative estimates for these operators attached to analytic functions on compact disks.

Now we define a new type of the complex q-Baskakov-Stancu operator
W n , q α , β ( f ) ( z ) = j = 0 [ n + j 1 ] ! [ n 1 ] ! q j ( j 1 ) 2 × [ [ α ] [ n ] + [ β ] , [ α ] + [ 1 ] [ n ] + [ β ] , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; f ] z j ( [ n ] + [ β ] ) j ,
(1.5)

where 0 α β ; for j = 0 , we take [ n ] [ n + 1 ] [ n + j 1 ] = 1 . We suppose that f is analytic on the disk | z | < R , R > 1 and has exponential growth in the compact disk with all derivatives bounded in [ 0 , ) by the same constant.

Note that taking α = β = 0 , W n , q α , β ( f ) ( z ) reduces to the complex q-Baskakov operator W n , q ( f ) ( z ) given in (1.4).

In this work, for such f and q > 1 , we study some approximation properties of the complex q-Baskakov-Stancu operator which is defined by forward differences.

2 Auxiliary results

In this section, we give some results which we shall use in the proof of theorems.

Lemma 1 Let us define e k ( z ) = z k , T n , k α , β ( z ) : = W n , q α , β ( e k ) ( z ) , and N 0 denotes the set of all nonnegative integers. Then, for all n , k N 0 , 0 α β and z C , we have the following recurrence formula:
T n , k + 1 α , β ( z ) = q z ( 1 + z q ) [ n ] + [ β ] D q T n , k α , β ( z q ) + [ n ] z + [ α ] ( [ n ] + [ β ] ) T n , k α , β ( z ) .
(2.1)
Hence
T n , 1 α , β ( z ) = [ n ] z + [ α ] [ n ] + [ β ] , T n , 2 α , β ( z ) = z ( 1 + z q ) [ n ] + [ β ] [ n ] [ n ] + [ β ] + ( [ n ] z + [ α ] [ n ] + [ β ] ) 2

for all z C .

Proof Now we can write
T n , k α , β ( z ) = j = 0 [ n ] [ n + 1 ] [ n + j 1 ] ( [ n ] + [ β ] ) j q j ( j 1 ) 2 × [ [ α ] ( [ n ] + [ β ] ) , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; e k ] z j .
(2.2)
Using relation (1.1) and taking f = e k , g = e 1 and x j = q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) , we obtain
[ [ α ] [ n ] + [ β ] , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; e k + 1 ] = q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) [ [ α ] [ n ] + [ β ] , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; e k ] + [ [ α ] [ n ] + [ β ] , , q j 2 [ α ] + [ j 1 ] q j 2 ( [ n ] + [ β ] ) ; e k ] ,
(2.3)
using this in T n , k + 1 α , β ( z ) we reach
T n , k + 1 α , β ( z ) = q z ( 1 + z q ) [ n ] + [ β ] D q T n , k α , β ( z q ) + [ n ] z + [ α ] [ n ] + [ β ] T n , k α , β ( z ) .

 □

Lemma 2 Let α and β satisfy 0 α β . Denoting e j ( z ) = z j and W n , q 0 , 0 ( e j ) by W n , q ( e j ) given in (1.4), for all n , k N 0 , we have the following recursive relation for the images of monomials e k under W n , q α , β in terms of W n , q ( e j ) , j = 0 , 1 , , k :
T n , k α , β ( z ) = j = 0 k ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j , z ) .
(2.4)
Proof We can use mathematical induction with respect to k. For k = 0 , equality (2.4) holds. Let it be true for k = m , namely
T n , m α , β ( z ) = j = 0 m ( m j ) [ n ] j [ α ] m j ( [ n ] + [ β ] ) m W n , q ( e j , z ) .
Using (2.1), we have
T n , m + 1 α , β ( z ) = q z ( 1 + z q ) [ n ] + [ β ] j = 0 m ( m j ) [ n ] j [ α ] m j ( [ n ] + [ β ] ) m D q W n , q ( e j , z q ) + [ n ] z + [ α ] [ n ] + [ β ] j = 0 m ( m j ) [ n ] j [ α ] m j ( [ n ] + [ β ] ) m W n , q ( e j , z ) = j = 0 m ( m j ) [ n ] j + 1 [ α ] m j ( [ n ] + [ β ] ) m + 1 × [ q z ( 1 + z q ) [ n ] D q W n , q ( e j , z q ) + [ n ] z + [ α ] [ n ] W n , q ( e j , z ) ] .
Taking into account the recurrence relation for the complex q-Baskakov operator in Lemma 2 in [13], we get
W n , q ( e j + 1 , z ) = q z ( 1 + z q ) [ n ] D q W n , q ( e j , z q ) + z W n , q ( e j , z ) ,
which implies
T n , m + 1 α , β ( z ) = j = 0 m ( m j ) [ n ] j + 1 [ α ] m j ( [ n ] + [ β ] ) m + 1 [ W n , q ( e j + 1 , z ) + [ α ] [ n ] W n , q ( e j , z ) ] = j = 1 m ( m j 1 ) [ n ] j [ α ] m j + 1 ( [ n ] + [ β ] ) m + 1 W n , q ( e j , z ) + j = 0 m ( m j ) [ n ] j [ α ] m j + 1 ( [ n ] + [ β ] ) m + 1 W n , q ( e j , z ) = j = 0 m + 1 ( m + 1 j ) [ n ] j [ α ] m j + 1 ( [ n ] + [ β ] ) m + 1 W n , q ( e j , z ) ,

which proves the lemma. □

3 Approximation by a complex q-Baskakov-Stancu operator

In this section, we give quantitative estimates concerning approximation with the following theorem.

Theorem 1 For 1 < R < , let
f : D ¯ R [ R , ) C

be a function with all its derivatives bounded in [ 0 , ) by the same positive constant, analytic in D R , namely f ( z ) = k = 0 c k z k for all z D R and suppose that there exist M > 0 and A ( 1 R , 1 ) , with the property | c k | M A k k ! for all k = 0 , 1 , (which implies | f ( z ) | M e A | z | for all z D R ).

Let 0 α β , q > 1 and 1 r < 1 A be arbitrary but fixed. Then, for all | z | r and n N , we have
| W n , q α , β ( f ) ( z ) f ( z ) | M 1 , r ( f ) [ n ] + [ β ] + [ β ] [ n ] + [ β ] M 2 , r ( f ) + [ α ] [ n ] + [ β ] M 3 , r ( f ) = M r , α , β ( f )
with
M 1 , r ( f ) = 6 k = 2 | c k | ( k + 1 ) ! ( k 1 ) r k < , M 2 , r ( f ) = k = 1 | c k | k r k < , M 3 , r ( f ) = k = 1 | c k | k r k 1 < .
Proof Using (2.1), one can obtain
T n , k α , β ( z ) z k = q z ( 1 + z q ) [ n ] + [ β ] D q ( T n , k 1 α , β ( z q ) ) + [ n ] z + [ α ] [ n ] + [ β ] ( T n , k 1 α , β ( z ) z k 1 ) + [ n ] z + [ α ] [ n ] + [ β ] z k 1 z k = z ( 1 + z q ) [ n ] + [ β ] q D q ( T n , k 1 α , β ( z q ) ) + [ n ] z + [ α ] [ n ] + [ β ] ( T n , k 1 α , β ( z ) z k 1 ) + ( [ n ] [ n ] + [ β ] 1 ) z k + [ α ] [ n ] + [ β ] z k 1 .
Moreover, we have
q D q ( T n , k 1 α , β ( z q ) ) = | D q ( T n , k 1 α , β ( w ) ) | w = z q .
(3.1)
Now from (3.1) and the Bernstein inequality (see [1]), we have
q D q ( T n , k 1 α , β ( z q ) ) = | D q ( T n , k 1 α , β ( z ) ) | | T n , k 1 α , β ( z ) | k 1 r T n , k 1 α , β r ,
where r is the standard maximum norm over D r = { z C : | z | r } . Passing to modulus for all | z | r and n N , we have that
| T n , k α , β ( z ) z k | r ( 1 + r ) [ n ] + [ β ] ( k 1 r ) T n , k 1 α , β r + [ n ] r + [ α ] [ n ] + [ β ] | T n , k 1 α , β ( z ) z k 1 | + ( [ n ] [ n ] + [ β ] 1 ) r k + [ α ] [ n ] + [ β ] r k 1 .
(3.2)
In order to get an estimate for T n , k 1 α , β r in (3.2), we use the following fact:
T n , k α , β ( z ) = j = 0 k [ n ] [ n + 1 ] [ n + j 1 ] ( [ n ] + [ β ] ) j q j ( j 1 ) 2 [ [ α ] [ n ] + [ β ] , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; e k ] z j
for k N . Taking into account Lemma 1 in [13] for q > 1 , | z | r , r 1 and (1.3), we have
T n , k α , β ( z ) r r j j = 0 k [ n ] [ n + 1 ] [ n + j 1 ] [ n ] j q j ( j 1 ) 2 × [ [ α ] [ n ] + [ β ] , , q j 1 [ α ] + [ j ] q j 1 ( [ n ] + [ β ] ) ; e k ] j = 0 k j ! k k 1 k j + 1 j ! r k j r j = r k j = 0 k k k 1 k j + 1 r k ( k + 1 ) ! .
(3.3)
Now, considering (3.3) in (3.2), for all | z | r , r 1 , with q > 1 and 0 α β ,
| T n , k α , β ( z ) z k | r ( 1 + r ) [ n ] + [ β ] r k 2 ( k + 1 ) ! + [ n ] r + [ α ] [ n ] + [ β ] | T n , k 1 α , β ( z ) z k 1 | + ( [ n ] [ n ] + [ β ] 1 ) r k + [ α ] [ n ] + [ β ] r k 1 [ n ] r + [ α ] [ n ] + [ β ] | T n , k 1 α , β ( z ) z k 1 | + r ( 1 + r ) [ n ] + [ β ] r k 2 ( k + 1 ) ! + [ β ] [ n ] + [ β ] r k + [ α ] [ n ] + [ β ] r k 1 r | T n , k 1 α , β ( z ) z k 1 | + 2 r k [ n ] + [ β ] ( k + 1 ) ! + [ β ] [ n ] + [ β ] r k + [ α ] [ n ] + [ β ] r k 1 .
(3.4)
Using the above inequalities beginning from k = 2 , 3 , and using the mathematical induction with respect to k, we arrive at
| T n , k α , β ( z ) z k | 2 r k [ n ] + [ β ] j = 2 k ( j + 1 ) ! + [ β ] [ n ] + [ β ] k r k + [ α ] [ n ] + [ β ] k r k 1 6 r k [ n ] + [ β ] ( k + 1 ) ! ( k 1 ) + [ β ] [ n ] + [ β ] k r k + [ α ] [ n ] + [ β ] k r k 1 .
(3.5)
Also we obtain the following: for k = 1 it is not difficult to see that
| T n , 1 α , β ( z ) z | = | [ α ] [ β ] z [ n ] + [ β ] | [ α ] + [ β ] r [ n ] + [ β ] .
Now, taking into account the proof of Theorem 1 in [13], we can write, for q > 1 , | z | r , r 1 , that
W n , q α , β ( f ) ( z ) = k = 0 c k T n , k α , β ( z ) ,
which implies
| W n , q α , β ( f ) ( z ) f ( z ) | k = 1 | c k | | T n , k α , β ( z ) z k | 6 [ n ] + [ β ] k = 1 | c k | ( k + 1 ) ! ( k 1 ) r k + [ β ] [ n ] + [ β ] k = 1 | c k | k r k + [ α ] [ n ] + [ β ] k = 1 | c k | k r k 1 = M 1 , r ( f ) [ n ] + [ β ] + [ β ] [ n ] + [ β ] M 2 , r ( f ) + [ α ] [ n ] + [ β ] M 3 , r ( f ) .
Here from the analyticity of f we have M 2 , r ( f ) < and M 3 , r ( f ) < . Also from the hypotheses of the theorem, one can get
M 1 , r ( f ) = 6 k = 1 | c k | ( k + 1 ) ! ( k 1 ) r k 6 M k = 1 ( k + 1 ) ( k 1 ) ( r A ) k

for all | z | r , 1 r 1 A , n N . □

Theorem 2 Let 0 α β , 1 r 1 A and q > 1 . Under the hypotheses of Theorem  1, for all | z | r and n N , the following Voronovskaja-type result
| W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) | K 1 , r ( f ) [ n ] 2 + j = 2 6 K j , r ( f ) ( [ n ] + [ β ] ) 2
holds with
K 1 , r ( f ) = 16 k = 3 | c k | ( k 1 ) ( k 2 ) 2 k ! r k < , K 2 , r ( f ) = [ α ] 2 k = 2 | c k | ( k 1 ) k ! 2 r k 2 < , K 3 , r ( f ) = 6 [ α ] k = 2 | c k | k 2 k ! r k 1 < , K 4 , r ( f ) = ( [ β ] 2 2 + 6 [ β ] ) k = 0 | c k | k 2 ( k + 1 ) ! r k < , K 5 , r ( f ) = [ α ] [ β ] k = 0 | c k | k ( k 1 ) r k 1 < , K 6 , r ( f ) = [ β ] 2 k = 0 | c k | k ( k 1 ) r k < .
Proof For all z D R , let us consider
W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) = W n , q ( f ) ( z ) f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) + W n , q α , β ( f ) ( z ) W n , q ( f ) ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) .
Using the fact that f ( z ) = k = 0 c k z k , we get
W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) = k = 2 c k ( W n , q ( e k ; z ) z k z 2 [ n ] ( 1 + z q ) k ( k 1 ) z k 2 ) + k = 2 c k ( T n , k α , β ( z ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 ) .
From Theorem 2 in [13], we have
| W n , q ( f ) ( z ) f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) | 16 [ n ] 2 k = 3 | c k | ( k 1 ) ( k 2 ) 2 k ! r k .
Furthermore, in order to estimate the second sum, using Lemma 2, we obtain
T n , k α , β ( z ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 = j = 0 k ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 = j = 0 k 1 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) + ( [ n ] k ( [ n ] + [ β ] ) k 1 ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 .
Also it is clear that
1 [ n ] k ( [ n ] + [ β ] ) k = j = 1 k 1 ( k j ) [ n ] j [ β ] k j ( [ n ] + [ β ] ) k j = 1 k 1 ( 1 [ n ] [ n ] + [ β ] ) = k [ β ] [ n ] + [ β ] ,
(3.6)
which implies
T n , k α , β ( z ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 = j = 0 k 2 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) + k [ n ] k 1 [ α ] ( [ n ] + [ β ] ) k W n , q ( e k 1 ; z ) j = 0 k 2 ( k j ) [ n ] j [ β ] k j ( [ n ] + [ β ] ) k W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 = j = 0 k 2 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) + k [ n ] k 1 [ α ] ( [ n ] + [ β ] ) k ( W n , q ( e k 1 ; z ) z k 1 ) j = 0 k 2 ( k j ) [ n ] j [ β ] k j ( [ n ] + [ β ] ) k W n , q ( e k ; z ) k [ n ] k 1 [ β ] ( [ n ] + [ β ] ) k ( W n , q ( e k ; z ) z k ) + ( [ n ] k 1 ( [ n ] + [ β ] ) k 1 1 ) k [ α ] [ n ] + [ β ] z k 1 + ( 1 [ n ] k 1 ( [ n ] + [ β ] ) k 1 ) k [ β ] [ n ] + [ β ] z k .
(3.7)
Now from (3.3) we obtain
| j = 0 k 2 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) | j = 0 k 2 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k | W n , q ( e j ; z ) | = j = 0 k 2 k ( k 1 ) ( k j ) ( k j 1 ) ( k 2 j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k | W n , q ( e j ; z ) | k ( k 1 ) 2 [ α ] 2 ( [ n ] + [ β ] ) 2 r k 2 ( k 1 ) ! j = 0 k 2 ( k 2 j ) [ n ] j [ α ] k 2 j ( [ n ] + [ β ] ) k 2 k ( k 1 ) 2 [ α ] 2 ( [ n ] + [ β ] ) 2 r k 2 ( k 1 ) ! .
(3.8)
Also, we need to prove the following inequality:
j = 0 k 2 ( k 2 j ) [ n ] j [ α ] k 2 j ( [ n ] + [ β ] ) k 2 = j = 0 k 2 ( k 2 j ) [ n ] j ( [ n ] + [ β ] ) j [ α ] k 2 j ( [ n ] + [ β ] ) k 2 j = ( [ n ] + [ α ] [ n ] + [ β ] ) k 2 1 .
(3.9)
Moreover, taking α = β = 0 in Theorem 1, we have
| W n , q ( e k ; z ) z k | 6 [ n ] r k ( k + 1 ) ! ( k 1 ) .
(3.10)
Writing (3.8), (3.6), (3.9) and (3.10) in (3.7), we have
| T n , k α , β ( z ) W n , q ( e k ; z ) [ α ] [ β ] z [ n ] + [ β ] k z k 1 | | j = 0 k 2 ( k j ) [ n ] j [ α ] k j ( [ n ] + [ β ] ) k W n , q ( e j ; z ) | + k [ n ] k 1 [ α ] ( [ n ] + [ β ] ) k | W n , q ( e k 1 ; z ) z k 1 | + | j = 0 k 2 ( k j ) [ n ] j [ β ] k j ( [ n ] + [ β ] ) k W n , q ( e k ; z ) | + k [ n ] k 1 [ β ] ( [ n ] + [ β ] ) k | W n , q ( e k ; z ) z k | + | [ n ] k 1 ( [ n ] + [ β ] ) k 1 1 | k [ α ] [ n ] + [ β ] | z | k 1 + | 1 [ n ] k 1 ( [ n ] + [ β ] ) k 1 | k [ β ] [ n ] + [ β ] | z | k ( k 1 ) k ! 2 [ α ] 2 ( [ n ] + [ β ] ) 2 r k 2 + k [ n ] k 1 [ α ] ( [ n ] + [ β ] ) k 6 [ n ] r k 1 k ! ( k 2 ) + r k ( k + 1 ) ! j = 0 k 2 ( k j ) [ n ] j [ β ] k j ( [ n ] + [ β ] ) k + k [ n ] k 1 [ β ] ( [ n ] + [ β ] ) k 6 [ n ] r k ( k + 1 ) ! ( k 1 ) + k ( k 1 ) [ α ] [ β ] ( [ n ] + [ β ] ) 2 r k 1 + k ( k 1 ) [ β ] 2 ( [ n ] + [ β ] ) 2 r k ( k 1 ) k ! 2 [ α ] 2 ( [ n ] + [ β ] ) 2 r k 2 + 6 k 2 [ α ] ( [ n ] + [ β ] ) 2 r k 1 k ! + k 2 [ β ] 2 ( k + 1 ) ! 2 ( [ n ] + [ β ] ) 2 r k + 6 k 2 ( k + 1 ) ! [ β ] ( [ n ] + [ β ] ) 2 r k + k ( k 1 ) [ α ] [ β ] ( [ n ] + [ β ] ) 2 r k 1 + k ( k 1 ) [ β ] 2 ( [ n ] + [ β ] ) 2 r k ( k 1 ) k ! 2 [ α ] 2 ( [ n ] + [ β ] ) 2 r k 2 + 6 k 2 [ α ] ( [ n ] + [ β ] ) 2 r k 1 k ! + ( [ β ] 2 2 + 6 [ β ] ) k 2 ( k + 1 ) ! ( [ n ] + [ β ] ) 2 r k + k ( k 1 ) [ α ] [ β ] ( [ n ] + [ β ] ) 2 r k 1 + k ( k 1 ) [ β ] 2 ( [ n ] + [ β ] ) 2 r k .

Thus the proof is completed. □

Now, let us give a lower estimate for the exact degree in approximation by W n , q α , β .

Theorem 3 Suppose that q > 1 and suppose that the hypotheses on f and on the constants R, M, A in the statement of Theorem  1 hold, and let 1 r < R , 0 α β . If f is not a polynomial of degree ≤0, then the lower estimate
W n , q α , β ( f ) f r C r α , β ( f ) [ n ]

holds for all n, where the constant C r α , β ( f ) depends on f, α, β, q and r.

Proof For all | z | r and n N , we get
W n , q α , β ( f ) ( z ) f ( z ) = 1 [ n ] { [ n ] [ n ] + [ β ] ( [ α ] [ β ] z ) f ( z ) + z 2 ( 1 + z q ) f ( z ) + 1 [ n ] [ n ] 2 ( W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) ) } = 1 [ n ] { ( [ α ] [ β ] z ) f ( z ) + z 2 ( 1 + z q ) f ( z ) + 1 [ n ] [ n ] 2 ( W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) ) + 1 [ n ] [ n ] 2 ( z 2 [ n ] ( 1 + z q ) f ( z ) [ β ] ( [ α ] [ β ] z ) [ n ] ( [ n ] + [ β ] ) f ( z ) ) } .
We set E k , n ( z ) by
E k , n ( z ) : = W n , q α , β ( f ) ( z ) f ( z ) [ α ] [ β ] z [ n ] + [ β ] f ( z ) z 2 [ n ] ( 1 + z q ) f ( z ) [ β ] ( [ α ] [ β ] z ) [ n ] ( [ n ] + [ β ] ) f ( z ) .
(3.11)
Passing to the norm and using the inequality
F + G r | F r G r | F r G r ,
we get
W n , q α , β ( f ) f r 1 [ n ] ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r 1 [ n ] [ n ] 2 E k , n r .
Since f is not a polynomial of degree ≤0 in D R , we have
( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r > 0 .
It can also be seen in [[1], pp.75-76]. Now, from Theorem 2 it follows that
[ n ] 2 E k , n r [ n ] 2 W n , q α , β ( f ) f ( [ α ] [ β ] e 1 [ n ] + [ β ] ) f e 1 2 [ n ] ( 1 + e 1 q ) f r + [ β ] ( [ α ] [ β ] e 1 ) f r j = 1 6 M j , r ( f ) + [ β ] ( [ α ] + [ β ] r ) f r .
Since 1 [ n ] 0 as n , for q > 1 , there exists an n 0 depending on f, r, α, β and q such that for all n n 0 ,
1 [ n ] ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r 1 [ n ] [ n ] 2 E k , n r 1 2 ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r ,
which implies
W n , q α , β ( f ) f r 1 2 [ n ] ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r
for all n n 0 . Now, for n { 1 , , n 0 1 } , we have
W n , q α , β ( f ) f r A r ( f ) [ n ]
with
A r ( f ) = [ n ] W n , q α , β ( f ) f r > 0 ,
which finally implies
W n , q α , β ( f ) f r C r α , β ( f ) [ n ]
for all n n 0 with
C r α , β ( f ) = min { A r , 1 ( f ) , , A r , n 0 1 ( f ) , 1 2 ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f r } .

This proves the theorem. □

Combining now Theorem 3 with Theorem 1, we immediately get the following equivalence result.

Remark 1 Suppose that q > 1 , 0 α β and that the hypotheses on f and on the constants R, M, A in the statement of Theorem 1 hold, and let 1 r < 1 A be fixed. If f is not a polynomial of degree ≤0, then we have the following equivalence:
W n , q α , β ( f ) f r 1 [ n ]

for all n, where the constants in the equivalence depend on f, α, β, q and r.

Concerning the approximation by the derivatives of complex q-Baskakov-Stancu operators, we can state the following theorem.

Theorem 4 Suppose that q > 1 and that the hypotheses on f and on the constants R, M, A in the statement of Theorem  1 hold, and let 0 α β , 1 r < r 1 < 1 A and p N be fixed. If f is not a polynomial of degree p 1 , then we have the following equivalence:
[ W n , q α , β ( f ) ] ( p ) f ( p ) r 1 [ n ]

for all n, where the constants in the equivalence depend on f (that is, on M, A), r, r 1 q and p.

Proof Denote by Γ the circle of radius r 1 with 1 r < r 1 < 1 A centered at 0. Since | z | r and γ Γ , we have | γ z | r 1 r and from Cauchy’s formulas and Theorem 1 we obtain, for all | z | r and n N , that
| [ W n , q α , β ( f , z ) ] ( p ) f ( p ) ( z ) | p ! 2 π | Γ W n , q α , β f ( γ ) f ( γ ) ( γ z ) p + 1 d γ | M r 1 , α , β ( f ) [ n ] p ! 2 π 2 π r 1 ( r 1 r ) p + 1 = M r 1 , α , β ( f ) [ n ] p ! r 1 ( r 1 r ) p + 1 ,

which proves one of the inequalities in the equivalence.

Now we need to prove the lower estimate. From Cauchy’s formula we get
[ W n , q α , β ( f , z ) ] ( p ) f ( p ) ( z ) = p ! 2 π i Γ W n , q α , β f ( γ ) f ( γ ) ( γ z ) p + 1 d γ .
Furthermore, using (3.11) one can have
W n , q α , β f ( γ ) f ( γ ) = 1 [ n ] { ( [ α ] [ β ] γ ) f ( γ ) + γ 2 ( 1 + γ q ) f ( γ ) + [ n ] 2 E k , n ( γ ) }
for all γ Γ and n N . Applications of Cauchy’s formula imply
[ W n , q α , β ( f , z ) ] ( p ) f ( p ) ( z ) = { 1 [ n ] p ! 2 π i Γ ( [ α ] [ β ] γ ) f ( γ ) + γ 2 ( 1 + γ q ) f ( γ ) ( γ z ) p + 1 d γ + 1 [ n ] p ! 2 π i Γ [ n ] 2 E k , n ( γ ) ( γ z ) p + 1 d γ } = 1 [ n ] { [ ( [ α ] [ β ] γ ) f ( γ ) + z 2 ( 1 + z q ) f ( z ) ] ( p ) + p ! 2 π i Γ [ n ] 2 E k , n ( γ ) ( γ z ) p + 1 d γ } .
Now passing to the norm r we obtain
[ W n , q α , β ( f ) ] ( p ) f ( p ) r 1 [ n ] { [ ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f ] ( p ) r 1 [ n ] p ! 2 π Γ [ n ] 2 E k , n ( γ ) ( γ z ) p + 1 d γ r } ,
and from Theorem 2 we have
p ! 2 π Γ [ n ] 2 E k , n ( γ ) ( γ z ) p + 1 d γ r p ! 2 π 2 π r 1 ( r 1 r ) p + 1 [ n ] 2 E k , n r 1 K 1 , r 1 ( f ) + [ n ] 2 j = 2 6 K j , r 1 ( f ) ( [ n ] + [ β ] ) 2 + [ β ] ( [ α ] + [ β ] r 1 ) f r 1 .
Since f is not a polynomial of degree ≤0 in D R , we have
[ ( [ α ] [ β ] e 1 ) f + e 1 2 ( 1 + e 1 q ) f ] ( p ) r > 0

(see [[1], pp.77-78]). The rest of the proof is obtained similarly to that of Theorem 3. □

Remark 2 Note that if we take α = β = 0 , then Theorems 1, 2, 3 and 4 reduce to the results in [13].

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Ankara University, Tandoğan
(2)
Department of Mathematics, Faculty of Arts and Science, Kırıkkale University

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© Özden and Arı; licensee Springer. 2014

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