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Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

  • Huo Tang EMAIL logo , Kaliappan Vijaya , Gangadharan Murugusundaramoorthy and Srikandan Sivasubramanian
Published/Copyright: May 20, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

Let f k ( z ) = z + ∑ n = 2 k a n z n be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } , Re { f k ( z ) / f ( z ) } , Re { f ′ ( z ) / f k ′ ( z ) } and Re { f k ′ ( z ) / f ′ ( z ) } , where f ( z ) belongs to the subclass J p , q m ( μ , α , β ) of analytic functions, defined by Sălăgean ( p , q ) -differential operator. In addition, the inclusion relations involving N δ ( e ) of this generalized function class are considered.

Keywords: analytic; univalent; (p, q)-differential operator; partial sum; inclusion relation
MSC 2010: 30C45; 30C50

1 Introduction and preliminaries

Let A denote the class of functions of the form

(1.1) f ( z ) = z + ∑ n = 2 ∞ a n z n ,

which are analytic and univalent in the open disc D = { z : ∣ z ∣ < 1 } . We also denote T a subclass of A introduced and studied by Silverman [1], consisting of functions of the form:

(1.2) f ( z ) = z − ∑ n = 2 ∞ a n z n , a n ≥ 0 ; z ∈ D .

For functions f ∈ A given by (1.1) and g ∈ A given by g ( z ) = z + ∑ n = 2 ∞ b n z n , we define the Hadamard product (or convolution) of f and g by ( f ∗ g ) ( z ) = z + ∑ n = 2 ∞ a n b n z n , z ∈ D .

We briefly recall here the notion of q-operators i.e. q-difference operator that plays vital role in the theory of hypergeometric series, quantum physics and in the operator theory. The application of q-calculus was initiated by Jackson [2] (also see [3–5]). Kanas and Răducanu [4] have used the fractional q-calculus operators in investigations of certain classes of functions which are analytic in D . For p > 0 , q > 0 the (p; q)-differential operator of a function f , analytic in D is defined for a function f ∈ A is, by definition, given as follows [2]:

(1.3) D p , q f ( z ) = f ( p z ) − f ( q z ) ( p − q ) z ( p ≠ q ) .

From (1.3), we have

(1.4) D p , q f ( z ) = 1 + ∑ n = 2 ∞ [ n ] p , q a n z n − 1 ,

where

(1.5) [ n ] p , q = p n − q n p − q , [ 0 ] p , q = 0 .

The twin-basic number is a natural generalization of q -number, that is

(1.6) [ n ] q = 1 − q n 1 − q , ( q ≠ 1 ) ,

which is sometimes called the basic number n .

One can easily verify that D p , q f ( z ) → f ′ ( z ) as p → 1 − and q → 1 − . It is clear that q -integers and ( p ; q ) -integers differ, that is, we cannot obtain ( p ; q ) -integers just by replacing q by q p in the definition of q -integers. However, (1.5) reduces to (1.6) for the case p = 1 . Thus, we can say that ( p ; q ) -calculus can be taken as a generalization of q -calculus. The ( p ; q ) -factorial is defined by

(1.7) [ n ] p , q ! = ∏ k = 1 n [ k ] p , q ! ( n ≥ 1 ) , [ 0 ] p , q ! = 1 .

Note that p → 1 − the ( p ; q ) -factorial reduces to the q -factorial. Also, clearly

lim p → 1 − lim q → 1 − = [ n ] p , q ! = n ! and lim p → 1 − lim a → 1 − = [ n ] p , q = n .

For details on q-calculus and (p, q)-calculus, one can refer to [2,6,7] and also references cited therein. Recently for f ∈ A , Govindaraj and Sivasubramanian [8] defined Sălăgean (q)-differential operator and further Kanas and Răducanu [4] defined and discussed Sălăgean (p, q)-differential operator as follows:

(1.8) D p , q 0 f ( z ) = f ( z ) , D p , q 1 f ( z ) = z D p , q f ( z ) , D p , q m f ( z ) = z D p , q m ( D p , q m − 1 f ( z ) ) , D p , q m f ( z ) = z + ∑ n = 2 ∞ [ n ] p , q m a n z n ( m ∈ N 0 , z ∈ D ) .

It is interesting that one can observe

(1.9) lim ( p , q ) → ( 1 − , 1 − ) D p , q m f ( z ) = D m f ( z ) = z + ∑ n = 2 ∞ n m a n z n ( m ∈ N 0 , z ∈ D ) ,

the familiar Sălăgean derivative [9].

For 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 , we let J p , q m ( μ , α , β ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.10) ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − α > β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 , z ∈ D ,

where D p , q m f ( z ) is given by (1.8). We further let TJ p , q m ( μ , α , β ) = J p , q m ( μ , α , β ) ∩ T .

By taking μ = 1 we get TJ p , q m ( 1 , α , β ) ≡ TSP p , q m ( α , β ) studied by Kanas and Răducanu [4]. Further by specializing the parameter μ = 0 , we define the following new subclass:

Remark 1.1

For μ = 0 , m ∈ N 0 , 0 ≤ α < 1 and β ≥ 0 , let TJ p , q m ( 0 , α , β ) ≡ USD p , q m ( α , β ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.11) ℜ D p , q m + 1 f ( z ) z − α > β D p , q m + 1 f ( z ) z − 1 , z ∈ D ,

where D p , q m f ( z ) is given by (1.8).

Remark 1.2

For μ = 0 , β = 0 , 0 ≤ α < 1 and m ∈ N 0 , let TJ p , q m ( 1 , α , 0 ) ≡ ℛ p , q m ( α ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.12) ℜ D p , q m + 1 f ( z ) z > α z ∈ D ,

where D p , q m f ( z ) is given by (1.8).

Remark 1.3

As lim ( p , q ) → ( 1 − , 1 − ) , D p , q m f ( z ) reduces to various interesting subclasses (as mentioned earlier) satisfying the analytic criterion

(1.13) ℜ D m + 1 f ( z ) ( 1 − μ ) z + μ D m f ( z ) − α > β D m f ( z ) ( 1 − μ ) z + μ D m f ( z ) − 1 , z ∈ D ,

where D m f ( z ) is given by (1.9).

2 Basic properties

In this section, we obtain the characterization properties for the classes TJ p , q m ( μ , α , β ) .

Theorem 2.1

A function f ( z ) of the form (1.1) is in J p , q m ( μ , α , β ) if

(2.1) ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) ∣ a n ∣ ≤ 1 − α ,

where 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 .

The result is sharp for the function

f n ( z ) = z − 1 − α [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) z n .

Proof

It suffices to show that

β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 − ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ 1 − α .

We have

β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 − ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ ( 1 + β ) D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ ( 1 + β ) ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q − μ ) ∣ a n ∣ ∣ z ∣ n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ ∣ a n ∣ ∣ z ∣ n − 1 .

As ∣ z ∣ → 1 − , the last expression is bounded above by 1 − α if (2.1) holds. It is obvious that the function f n satisfies the inequality (2.1), and thus 1 − α cannot be replaced by a larger number. Therefore, we need only to prove that f ∈ TJ p , q m ( μ , α , β ) . Since

ℜ 1 − ∑ n = 2 ∞ [ n ] p , q m + 1 a n z n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ a n z n − 1 − α > β ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q − μ ) a n z n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ a n z n − 1 .

Letting z → 1 along the real axis, we obtain the desired inequality given in (2.1).□

Corollary 2.2

If f ∈ TJ p , q m ( μ , α , β ) , then

(2.2) ∣ a n ∣ ≤ 1 − α Φ p , q m ( μ , α , β ) .

Equality holds for the function f ( z ) = z − 1 − α Φ p , q m ( μ , α , β ) z n , where

Φ p , q m ( μ , α , β ) = [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) .

Throughout this paper for convenience, unless otherwise stated, we let

(2.3) Φ n m = Φ p , q m ( μ , α , β ) = [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) )

and

(2.4) Φ 2 m = Φ p , q m ( μ , α , β ) = [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ,

where 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 .

3 Partial sums

Silverman [10] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. In this section, following the earlier work by Silverman [10] and also the work cited in [11–15] on partial sums of analytic functions, we study the ratio of a function of the form (1.1) to its sequence of partial sums of the form

f k ( z ) = z + ∑ n = 2 k a n z n ,

when the coefficients of f ( z ) satisfy the condition (2.1).

Theorem 3.1

If f ∈ A of the form (1.1) satisfies the condition (2.1), then

(3.1) ℜ f ( z ) f k ( z ) ≥ Φ k + 1 m − 1 + α Φ k + 1 m ( z ∈ D ) ,

where

(3.2) Φ n m = Φ p , q m ( μ , α , β ) ≥ 1 − α , i f n = 2 , 3 , … , k , Φ k + 1 m , i f n = k + 1 , k + 2 , … .

The result (3.1) is sharp with the function given by

(3.3) f ( z ) = z + 1 − α Φ k + 1 m z k + 1 .

Proof

Define the function w ( z ) by

(3.4) 1 + w ( z ) 1 − w ( z ) = Φ k + 1 m 1 − α f ( z ) f k ( z ) − Φ k + 1 m − 1 + α Φ k + 1 m = 1 + ∑ n = 2 k a n z n − 1 + Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 1 + ∑ n = 2 k a n z n − 1 .

It suffices to show that ∣ w ( z ) ∣ ≤ 1 . Now, from (3.4) we can write

w ( z ) = Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 2 + 2 ∑ n = 2 k a n z n − 1 + Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 .

Hence, we obtain

∣ w ( z ) ∣ ≤ Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ 2 − 2 ∑ n = 2 k ∣ a n ∣ − Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ .

Now ∣ w ( z ) ∣ ≤ 1 if and only if

2 Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ ≤ 2 − 2 ∑ n = 2 k ∣ a n ∣ ,

or, equivalently,

∑ n = 2 k ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k + 1 m 1 − α ∣ a n ∣ ≤ 1 .

From the condition (2.1), it is sufficient to show that

∑ n = 2 k ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k + 1 m 1 − α ∣ a n ∣ ≤ ∑ n = 2 ∞ Φ n m 1 − α ∣ a n ∣ ,

which is equivalent to

(3.5) ∑ n = 2 k Φ k m − 1 + α 1 − α ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k m − Φ k + 1 m 1 − α ∣ a n ∣ ≥ 0 .

To see that the function given by (3.3) gives the sharp result, we observe that for z = r e i π / n

f ( z ) f k ( z ) = 1 + 1 − α Φ k + 1 m z n → 1 − 1 − α Φ k + 1 m = Φ k + 1 m − 1 + α Φ k + 1 m , when r → 1 − . □

Theorem 3.2

If f of the form (1.1) satisfies the condition (2.1), then

(3.6) ℜ f k ( z ) f ( z ) ≥ Φ k + 1 m Φ k + 1 m + 1 − α ( z ∈ D ) ,

where Φ n + 1 m ≥ 1 − α and

(3.7) Φ k m ≥ 1 − α , i f n = 2 , 3 , … , k , Φ k + 1 m , i f n = k + 1 , k + 2 , … .

The result (3.6) is sharp with the function given by (3.3).

Proof

The proof follows by defining

1 + w ( z ) 1 − w ( z ) = Φ k + 1 m + 1 − α 1 − α f k ( z ) f ( z ) − Φ k + 1 m Φ k + 1 m + 1 − α

and much akin to similar arguments in Theorem 3.1.□

We next turn to ratios involving derivatives.

Theorem 3.3

If f of the form (1.1) satisfies the condition (2.1), then

(3.8) ℜ f ′ ( z ) f n ′ ( z ) ≥ Φ n + 1 m − ( n + 1 ) ( 1 − α ) Φ n + 1 m ( z ∈ D )

and

(3.9) ℜ f n ′ ( z ) f ′ ( z ) ≥ Φ n + 1 m Φ n + 1 m + ( n + 1 ) ( 1 − γ ) ( z ∈ D ) ,

where Φ n + 1 m ≥ ( n + 1 ) ( 1 − α ) and

(3.10) Φ k m ≥ k ( 1 − α ) , i f k = 2 , 3 , … , n , k Φ n + 1 m n + 1 , i f k = n + 1 , n + 2 , … .

The results are sharp with the function given by (3.3).

Proof

We write

1 + w ( z ) 1 − w ( z ) = Φ n + 1 m ( n + 1 ) ( 1 − α ) f ′ ( z ) f n ′ ( z ) − Φ n + 1 m − ( n + 1 ) ( 1 − α ) Φ n + 1 m ,

where

w ( z ) = Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k a k z k − 1 2 + 2 ∑ k = 2 n k a k z k − 1 + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k a k z k − 1 .

Now ∣ w ( z ) ∣ ≤ 1 if and only if

∑ k = 2 n k ∣ a k ∣ + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k ∣ a k ∣ ≤ 1 .

From the condition (2.1), it is sufficient to show that

∑ k = 2 n k ∣ a k ∣ + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k ∣ a k ∣ ≤ ∑ k = 2 ∞ Φ k m 1 − γ ∣ a k ∣ ,

which is equivalent to

∑ k = 2 n Φ k m − ( 1 − α ) k 1 − α ∣ a k ∣ + ∑ k = n + 1 ∞ ( n + 1 ) Φ k m − k Φ n + 1 m ( n + 1 ) ( 1 − α ) ∣ a k ∣ ≥ 0 .

To prove the result (3.9), we define the function w ( z ) by

1 + w ( z ) 1 − w ( z ) = ( n + 1 ) ( 1 − α ) + Φ n + 1 m ( 1 − α ) ( n + 1 ) f n ′ ( z ) f ′ ( z ) − Φ n + 1 m ( n + 1 ) ( 1 − α ) + Φ n + 1 m

and by similar arguments in the first part we get desired result.□

4 Inclusion relations involving N δ ( e )

In this section following [16–18], we define the n , δ neighborhood of function f ( z ) ∈ T and discuss the inclusion relations involving N δ ( e ) .

(4.1) N δ ( f ) = g ∈ T : g ( z ) = z − ∑ n = 2 ∞ b n z n and ∑ n = 2 ∞ n ∣ a n − b n ∣ ≤ δ .

Particularly for the identity function e ( z ) = z , we have

(4.2) N δ ( e ) = g ∈ T : g ( z ) = z − ∑ n = 2 ∞ b n z n and ∑ n = 2 ∞ n ∣ b n ∣ ≤ δ .

Theorem 4.1

Let

(4.3) δ = 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

Then TJ p , q m ( μ , α , β ) ⊂ N δ ( e ) .

Proof

For f ∈ TJ p , q m ( μ , α , β ) , Theorem 2.1, yields

[ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ∑ n = 2 ∞ a n ≤ 1 − α ,

so that

(4.4) ∑ n = 2 ∞ a n ≤ 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

On the other hand, from (2.1) and (4.4) we have

(4.5) [ 2 ] p , q m ( 1 + β ) ∑ n = 2 ∞ [ n ] p , q a n ≤ 1 − α + [ 2 ] p , q m μ ( α + β ) ∑ n = 2 ∞ a n ≤ 1 − α + [ 2 ] p , q m μ ( α + β ) ( 1 − α ) [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ≤ [ 2 ] p , q m ( 1 + β ) ( 1 − α ) [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ∑ n = 2 ∞ [ n ] p , q a n ≤ 1 − α [ 2 ] p , q ( 1 + β ) − μ ( α + β ) .

□

Now we determine the neighborhood for each of the class TJ p , q m ( μ , α , β ) , which we define as follows.

A function f ∈ T is said to be in the class TJ p , q m ( μ , α , β ) if there exists a function g ∈ TJ m ( μ , α , β , η ) such that

(4.6) f ( z ) g ( z ) − 1 < 1 − η , ( z ∈ D , 0 ≤ η < 1 ) .

Theorem 4.2

If g ∈ TJ p , q m ( μ , α , β ) and

(4.7) η = 1 − δ [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) 2 [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

Then N δ ( g ) ⊂ TJ p , q m ( μ , α , β , η ) .

Proof

Suppose that f ∈ N δ ( g ) , then we find from 4.1 that

∑ n = 2 ∞ n ∣ a n − b n ∣ ≤ δ ,

which implies that the coefficient inequality

∑ n = 2 ∞ ∣ a n − b n ∣ ≤ δ 2 .

Next, since g ∈ TJ p , q m ( μ , α , β ) , we have

∑ n = 2 ∞ b n ≤ 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

So that

f ( z ) g ( z ) − 1 < ∑ n = 2 ∞ ∣ a n − b n ∣ 1 − ∑ n = 2 ∞ b n ≤ δ 2 × [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) − ( 1 − α ) ≤ 1 − η ,

provided that η is given precisely by (4.7). Thus by definition, f ∈ TJ p , q m ( μ , α , β , η ) for η given by (4.7), which completes the proof.□

5 Concluding remarks and observations

As a special case of the aforementioned theorems, we can determine new sharp lower bounds for ℜ f ( z ) f n ( z ) , ℜ f n ( z ) f ( z ) , ℜ f ′ ( z ) f n ′ ( z ) and ℜ f n ′ ( z ) f ′ ( z ) for various function classes stated in Remarks 1.1 and 1.2 and upon specializing the values of μ and β one can deduce various new subclasses on p , q -difference operator and prove the above partial sums and neighborhood results.

  1. Funding information: The research of Huo Tang was partly supported by the Natural Science Foundation of the People’s Republic of China under Grants 11561001 and 11271045, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People’s Republic of China under Grant 2018MS01026 and the Natural Science Foundation of Chifeng of Inner Mongolia. The work of S. Sivasubramanian was supported by a grant from the Science and Engineering Research Board, Government of India under Mathematical Research Impact Centric Support of Department of Science and Technology (DST) (vide ref: MTR/2017/000607).

  2. Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-07-09
Revised: 2021-01-07
Accepted: 2021-01-07
Published Online: 2021-05-20

© 2021 Huo Tang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0028
Keywords for this article
analytic; univalent; (p, q)-differential operator; partial sum; inclusion relation
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  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
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  26. Hodge-Deligne polynomials of character varieties of free abelian groups
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  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
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  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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