Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
-
Zeynep Kayar
and
Billur Kaymakçalan
Abstract
Reverse Pachpatte-type inequalities are concave generalizations of the well-known Bennett-Leindler-type inequalities. We establish reverse nabla Pachpatte-type dynamic inequalities taking account of concavity. It is the first time that converses of Pachpatte-type inequalities are obtained in the nabla time scale calculus as well as for its special cases such as continuous and discrete cases and for the dual results obtained in the delta time scale calculus. Moreover, some of our results extend the related ones when concavity has been removed.
1 Introduction
Our aim is to obtain converses of Pachpatte-type dynamic inequalities in the nabla time scale calculus, which are concave generalizations of nabla Bennett-Leindler-type dynamic inequalities.
Since Bennett-Leindler-type inequalities turn information about derivatives of functions into information about the size of the function, they are essential part of all areas of mathematics and useful in various applications, see [1–4].
The Bennett-Leindler inequality has appeared as being converse of celebrated Hardy’s inequality.
Hardy [5] obtained the following discrete inequality for a sequence
and then Hardy et al. [6, Theorem 330] derived continuous version of inequality (1) as
where
Although a great deal of attention has been paid to Hardy’s inequalities to improve, generalize, and prove them in different ways and there are numerous research papers and also some books, we restrict ourselves to focus on the direction of Copson [7, Theorems 1.1, 2.1] and [8, Theorems 1, 3] for the generalization. In this direction, we refer the books [1,2,6,9,10] and the articles [7,11–16] and [8,17–20].
The convex extensions of Hardy-Copson-type inequalities have showed up in the literature after the celebrated papers of Pachpatte. Pachpatte [21] and Hwang and Yang [22] generalized the discrete Hardy-Copson inequalities as
where
Then, continuous version of discrete Pachpatte’s inequality (3) was derived in 1990 by Pachpatte [23] as follows: If
where
The following inequality was established by Pachpatte [24] in 1994, which is another continuous variant of his discrete inequality (3). If
where the functions
After that Pečarić and Hanjš [25] made a combination of inequalities (4) and (5) as
where the functions
The Bennett-Leindler inequality, which is a converse of the Hardy-Copson inequality, first appeared in the literature when Hardy and Littlewood [26] obtained its discrete version in 1927. The most generalized discrete Bennett-Leindler inequalities were derived by Copson [7], Bennett [27], and Leindler [12] for nonnegative sequences
where
and for
As far as we know, Hardy et al. [6, Theorem 337] established the first reverse version of the continuous Hardy-Copson inequality (2) for
where
Below is the last continuous Bennett-Leindler inequalities, which are continuous counterparts of the discrete Bennett-Leindler inequalities (7) and (8), obtained by Copson in [8, Theorems 4, 2], respectively,
and
where the functions
Several renowned inequalities have been developed in arbitrary time scales both in the delta time scale calculus [28–30] and in the nabla case [31–37] owing to the discovery of the calculus on a time scale [38–42].
Dynamic Hardy-Copson-type inequalities have been initially obtained by the delta approach. These unifications are presented in the book [43] and in the articles [44–52]. Their nabla analogues are given in [53–55].
The reverse delta Hardy-Copson-type inequalities, named delta Bennett-Leindler inequalities, are shown in [49,56–58] for
Some Pachpatte-type dynamic inequalities, which are convex generalizations of Hardy-Copson-type dynamic inequalities, were obtained in [48,59,60] via delta time scale calculus whereas their nabla counterparts can be found in [61,62].
Although dynamic Hardy-Copson-type inequalities were generalized via convexity in the time scale calculus by using delta and nabla approaches, their converse inequalities, which are named Bennett-Leindler dynamic inequalities, have not been considered so far. Therefore, our aim is to establish concave generalizations of the aforementioned Bennett-Leindler-type dynamic inequalities and to establish converses of nabla Pachpatte-type inequalities, which do not exist in the literature. We derive novel results for the nabla case as well as for the delta, continuous and discrete cases.
2 Preliminaries
This section is devoted to present the main definitions and theorems of the nabla time scale calculus. The fundamental theories of the delta and nabla calculi can be found in the books [38,39] or in the articles [40–42,48,54].
If
The
for all
The nabla derivative satisfies the following.
Lemma 2.1
[38,41] Let
If f is continuous at a left scattered point t, then f is nabla differentiable at t with
If f is nabla differentiable at t, then
A function
If
Some of the properties of the nabla integral are gathered next.
Lemma 2.2
[38,41] Let
Integration by parts formula holds:
Lemma 2.3
(Chain rule for the nabla derivative) [33] If
Lemma 2.4
(Hölder’s inequality) [32] For
When
is satisfied.
The reverse nabla Jensen’s inequality is a main tool in our results.
Lemma 2.5
(Jensen’s inequality) [32,63] For
holds.
If
holds.
3 Converses of Pachpatte-type inequalities
The following auxiliary functions will play central roles in our results:
where
Since converses of Pachpatte-type dynamic inequalities are concave generalizations of nabla Bennett-Leindler-type dynamic inequalities, in the proofs of the theorems we first employ concavity and subsequently prove Bennett-Leindler-type dynamic inequalities.
Theorem 3.1
For the functions
for
- (12)
- (13)
for
- (14)
- (15)
Proof
We first prove (I). Concavity of the function
(I)–(i) After taking account of inequality (16), estimation of the left-hand side of inequality (12) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (17) and obtain
where
For
where we have used
Moreover, for
where we have used
Inequality (20) implies that
and
Then, one can obtain from inequality (21) that
Then, inequality (18) reduces to
The reverse Hölder inequality (9) for the constants
Taking
(I)–(ii) After taking account of inequality (16), estimation of the left-hand side of inequality (13) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (23) and obtain
where
Then, one can obtain from inequality (22) that
Then, substituting the above inequality and inequality (19) to inequality (24) provides
The reverse Hölder inequality (9) for the constants
Now, we prove (II). Concavity of the function
(II)–(i) After taking account of inequality (25), estimation of the left-hand side of inequality (14) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (26) and obtain
where
For
where we have used
Then, one can obtain from inequality (21) that
Then, inequality (27) reduces to
The reverse Hölder inequality (9) for the constants
(II)–(ii) After taking account of inequality (25), estimation of the left-hand side of inequality (15) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (29) and obtain
where
Then, one can obtain from inequality (22) that
Then, substituting the above inequality and inequality (28) to inequality (30) provides
where
Remark 3.2
When
Corollary 3.3
Delta versions of the nabla inequalities (12)–(15) can be derived for the following functions:
Let
Then, nabla inequalities (12)–(15) turn into new delta inequalities. For a concave function C and for the functions X, V, and Y defined as in (31) and
for
- (32)
- (33)
for
- (34)
- (35)
Theorem 3.4
For the functions
for
- (36)
- (37)
for
- (38)
- (39)
Proof
Using the same structure as suggested in the proof of Theorem 3.1 leads to the proof.
(I)–(i) After employing concavity of the function
For
where we have used
Inequality (40) implies that
and
By applying the similar ideas of the proof of Theorem 3.1 for the functions
The reverse Hölder inequality (9) for the constants
(I)–(ii) After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
(II)-(i) After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
(II)–(ii) After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
Remark 3.5
When
When
Corollary 3.6
Let
Then, nabla inequalities (36)–(39) turn into new delta inequalities. For a concave function C and for the functions
for
- (43)
- (44)
for
- (45)
- (46)
When
Theorem 3.7
For the functions
for
- (47)
- (48)
for
- (49)
- (50)
Proof
We first prove (I). Concavity of the function
(I)–(i): After taking account of inequality (51), estimation of the left-hand side of inequality (47) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (52) and obtain
where
For
where we have used
Moreover, for
where
Inequality (55) implies that
and
Then, one can obtain from inequality (56) that
Then, inequality (53) reduces to
The reverse Hölder inequality (9) for the constants
Taking
(I)–(ii): After taking account of inequality (51), estimation of the left-hand side of inequality (48) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (59) and obtain
where
Then, one can obtain from inequality (57) that
Then, substituting the above inequality and inequality (54) to inequality (60) provides
The reverse Hölder inequality (9) for the constants
Now, we prove (II). Concavity of the function
(II)–(i) After taking account of inequality (61), estimation of the left-hand side of inequality (49) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (62) and obtain
where
For
where we have used
Then, one can obtain from inequality (56) that
Then, inequality (63) reduces to
The reverse Hölder inequality (9) for the constants
(II)–(ii): After taking account of inequality (61), estimation of the left-hand side of inequality (50) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (66) and obtain
where
Then, one can obtain from inequality (57) that
Then, substituting above inequality and inequality (64) to inequality (67) provides
The reverse Hölder inequality (9) for the constants
Remark 3.8
When
Corollary 3.9
Let
Then, nabla inequalities (47)–(50) turn into new delta inequalities. For a concave function C and for the functions
for
- (68)
- (69)
for
- (70)
- (71)
Theorem 3.10
For the functions
for
- (72)
- (73)
for
- (74)
- (75)
Proof
Using the same structure as suggested in the proof of Theorem 3.7 leads to the proof.
(I)–(i): After employing concavity of the function
For
where we have used
Inequality (76) implies that
and
By proceeding similar to the proof of Theorem 3.7 for the functions
The reverse Hölder inequality (9) for the constants
(I)–(ii): After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
(II)–(i) After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
(II)-(ii) After employing concavity of the function
Then, estimating the function
The reverse Hölder inequality (9) for the constants
Remark 3.11
When
When
Corollary 3.12
Let
Then, nabla inequalities (72)–(75) turn into new delta inequalities. For a concave function C and for the functions
for
- (78)
- (79)
for
- (80)
- (81)
When
Theorem 3.13
For the functions
for
- (82)
- (83)
for
- (84)
- (85)
Proof
We combine the proofs of Theorems 3.4 and 3.10.
(I)-(i): After taking account of inequality (51), estimation of the left-hand side of inequality (82) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (86) and obtain
where
For
where we have used
Inequality (88) implies that
and
Then, one can obtain from inequality (89) that
Then, combination of inequality (54) and inequality (91) in inequality (87) reduces to
The reverse Hölder inequality (9) implies inequality (82).
(I)-(ii): After taking account of inequality (51), estimation of the left-hand side of inequality (83) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (92) and obtain
where
Then, one can obtain from inequality (90) that
Then, substituting the above inequality and inequality (54) to inequality (93) provides
The reverse Hölder inequality (9) implies inequality (83).
Now, we prove (II).
(II)-(i): From inequality (61), the left-hand side of inequality (84) can be estimated as
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (94) and obtain
where
Then, one can obtain from inequality (89) that
Then, substituting the above inequality and inequality (64) to inequality (93) provides
The reverse Hölder inequality (9) implies inequality (84).
(II)-(ii): From inequality (61), the left-hand side of inequality (85) can be estimated as
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (66) and obtain
where
Then, one can obtain from inequality (90) that
Then, substituting the above inequality and inequality (64) to inequality (95) provides
Remark 3.14
When
When
Remark 3.15
Inequality (83) can also be obtained as follows:
Inequality (84) can also be obtained as follows:
Corollary 3.16
Let the constant
Then, nabla inequalities (82)–(85) turn into new delta inequalities. For a concave function C and for the functions
for
- (96)
- (97)
for
- (98)
- (99)
When
Theorem 3.17
For the functions
for
- (100)
- (101)
for
- (102)
- (103)
Proof
We combine the proofs of Theorems 3.4 and 3.10.
(I)-(i) After taking account of inequality (16), estimation of the left-hand side of inequality (100) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (104) and obtain
where
For
where
Inequality (106) implies that
and
Then, one can obtain from inequality (108) that
Then, combination of inequality (109) and inequality (19) in inequality (105) provides
The reverse Hölder inequality (9) implies inequality (100).
(I)-(ii) After taking account of inequality (16), estimation of the left-hand side of inequality (101) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (110) and obtain
where
Then, one can obtain from inequality (107) that
Then, combination of inequality (112) and inequality (19) in inequality (111) provides
The reverse Hölder inequality (9) implies inequality (101).
(II)-(i): After taking account of inequality (25), estimation of the left-hand side of inequality (102) implies
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (113) and obtain
where
Then, one can obtain from inequality (107) that
Then, substituting inequality (115) and inequality (28) to inequality (114) provides
The reverse Hölder inequality (9) implies inequality (102).
(II)-(ii): From inequality (25), the left-hand side of inequality (103) can be estimated as
In order to obtain nabla Bennett-Leindler-type inequalities, we apply integration by parts formula to the right-hand side of inequality (116) and obtain
where
Then, one can obtain from inequality (107) that
Then, substituting inequality (118) and inequality (28) to inequality (117) provides
The reverse Hölder inequality (9) implies inequality (103).□
Remark 3.18
When
When
Remark 3.19
Inequality (100) can also be obtained as follows:
Inequality (103) can also be obtained as follows.
Corollary 3.20
Let the constant
Then, nabla inequalities (100)–(103) turn into new delta inequalities. For a concave function C and for the functions
for
- (119)
- (120)
for
- (121)
- (122)
When
4 Conclusion
Since time scale calculus enables us to avoid the separate discussion of the two cases, which are continuous and discrete cases, the unification of these cases by nabla calculus has gained importance in recent years.
Moreover, convexity/concavity play key roles in functional analysis, optimization and control theory.
In this article, concave generalizations of nabla Bennett-Leindler-type inequalities, which are analagous to the results of the delta time scale calculus and are unifications of the related continuous and discrete results, were established. These novel nabla inequalities not only provided new delta, continuous and discrete results but also could serve as starting points for the new results in diamond alpha calculus, which is a linear combination of delta and nabla time scale calculi.
Acknowledgments
The authors are very grateful to the referees for the valuable comments and remarks which provide valuable insights and helped to polish the content of the article and improve its quality.
-
Funding information: The authors state no funding involved.
-
Author contributions: All authors accept responsibility for the entire content of this manuscript and approve its submission.
-
Conflict of interest: The author states no conflict of interest.
-
Ethical approval: The conducted research is not related to either human or animal use.
-
Data availability statement: No data is associated with this research.
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Articles in the same Issue
- Research Articles
- On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
- Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
- Grand Triebel-Lizorkin-Morrey spaces
- Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
- Proofs of two conjectures involving sums of normalized Narayana numbers
- On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
- New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
- Some inequalities for rational function with prescribed poles and restricted zeros
- Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
- Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
- Fixed point results for contractions of polynomial type
- Estimation for spatial semi-functional partial linear regression model with missing response at random
- Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
- New forms of bilateral inequalities for K-g-frames
- Rate of pole detection using Padé approximants to polynomial expansions
- Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
- Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
- Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
- New topologies derived from the old one via operators
- Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
- Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
- Nonparametric expectile shortfall regression for functional data
- Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
- A generalized p-Laplacian problem with parameters
- A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
- Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
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- Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
- Solutions of several general quadratic partial differential-difference equations in ℂ2
- Inequalities for the generalized trigonometric functions with respect to weighted power mean
- Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
- Hankel determinants for q-starlike functions connected with q-sine function
- System of partial differential hemivariational inequalities involving nonlocal boundary conditions
- A new family of multivalent functions defined by certain forms of the quantum integral operator
- A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
- Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
- Behavior of spatial curves under different transformations in Euclidean 4-space
- Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
- A new kind of Durrmeyer-Stancu-type operators
- A study of generalized Mittag-Leffler-type function of arbitrary order
- On the approximation of Kantorovich-type Szàsz-Charlier operators
- Split quaternion Fourier transforms for two-dimensional real invariant field
- Review Article
- Characterization generalized derivations of tensor products of nonassociative algebras
- Special Issue on Differential Equations and Numerical Analysis - Part II
- Existence and optimal control of Hilfer fractional evolution equations
- Persistence of a unique periodic wave train in convecting shallow water fluid
- Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
- Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
- Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
- Stability and bifurcation analysis of a modified chemostat model
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
- Analytic solutions of a generalized complex multi-dimensional system with fractional order
- Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
- Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
- On the sum form functional equation related to diversity index
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
- Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
- Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
- Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
