It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.
The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetric q-differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.
Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III).
A class of Briot–Bouquet differential equations is a magnificent part of investigating the geometric behaviors of analytic functions, using the subordination and superordination concepts. In this work, we aim to formulate a new differential operator with complex connections (coefficients) in the open unit disk and generalize a class of Briot–Bouquet differential equations (BBDEs). We study and generalize new classes of analytic functions based on the new differential operator. Consequently, we define a linear operator with applications.
Problem statement: Spatial modeling has many applications in various fields like agriculture, meteorology, forestry and it takes into consideration the spatial correlation structures. In the field of forestry the growth rate, in particular, the diameter of trees is usually an important parameter. The growth rate of trees in a forest is likely to be influenced by various factors like nutrients, fertility of soil, sunshine and rainfall. In this study, we investigated the spatial correlation of the mean diameter of trees in the natural Dipterocarp forest in Gunung Tebu forest reserve, Terengganu, Malaysia. Approach: The diameters were measured using the diameter tape and the unit of measurement is in centimeters (cm). The main sampling unit was 1 ha plot of 100 by 100 m located approximately in the centre of each treatment block. Within the 1 ha sample plot, the quadrants (20 by 20 m) were numbered consecutively from 1-25 and in the outer 16 quadrants; all trees having a diameter at breast height over bark (dbh) of 15.0 cm or more are individually numbered, tagged and enumerated. Using the rook's and queen's neighborhood structure, we computed the Moran's spatial correlation coefficient for the mean diameter of trees in each quadrant for the years 1975 up to 1986. Results: We found that there was a negative spatial correlation among the mean diameter of trees in the 16 quadrants (cases) of the natural Dipterocarp forest in Gunung Tebu forest reserve, Terengganu at α level 0.10. Conclusion/Recommendations: The existence of negative spatial correlation indicated that there was competition among the trees in Dipterocarp forest as a result of tree growth over time which was affected by species, size, age and other environmental factors. Further research will concentrate on the spatial modeling of diameter of trees for the years where negative correlation was found
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