Subclasses Of Multivalent Analytic Functions Research Articles

On certain subclasses of multivalent analytic functions

In the present paper the authors introduce two new subclasses of multivalent analytic functions. Distortion inequalities and partial sums of functions in these classes are given.

Inclusion and Neighborhood Properties for Certain Classes of Multivalently Analytic Functions

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the classes introduced here.

New Subclasses of Multivalent Analytic Functions Associated with a Linear Operator

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we consider two subclasses <svg style="vertical-align:-5.73167pt;width:113.5px;" id="M1" height="19.362499" version="1.1" viewBox="0 0 113.5 19.362499" width="113.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D439" d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5&#xA;t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z" /></g> <g transform="matrix(.012,-0,0,-.012,10.637,16.25)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g><g transform="matrix(.012,-0,0,-.012,17.693,16.25)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.012,-0,0,-.012,20.406,16.25)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> <g transform="matrix(.017,-0,0,-.017,27,12.162)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,32.882,12.162)"><path id="x1D44E" d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387&#xA;q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" /></g><g transform="matrix(.017,-0,0,-.017,41.483,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,48.18,12.162)"><path id="x1D450" d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" /></g><g transform="matrix(.017,-0,0,-.017,55.082,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,61.796,12.162)"><path id="x1D706" d="M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5&#xA;q27 -249 51 -392q7 -46 23 -46q24 0 70 60z" /></g><g transform="matrix(.017,-0,0,-.017,71.18,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,77.894,12.162)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g><g transform="matrix(.017,-0,0,-.017,90.354,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,97.052,12.162)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g><g transform="matrix(.017,-0,0,-.017,107.54,12.162)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> and <svg style="vertical-align:-5.73167pt;width:115.225px;" id="M2" height="19.362499" version="1.1" viewBox="0 0 115.225 19.362499" width="115.225" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g> <g transform="matrix(.012,-0,0,-.012,12.375,16.25)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.012,-0,0,-.012,19.431,16.25)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,22.144,16.25)"><use xlink:href="#x1D45B"/></g> <g transform="matrix(.017,-0,0,-.017,28.738,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,34.619,12.162)"><use xlink:href="#x1D44E"/></g><g transform="matrix(.017,-0,0,-.017,43.22,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,49.918,12.162)"><use xlink:href="#x1D450"/></g><g transform="matrix(.017,-0,0,-.017,56.819,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,63.534,12.162)"><use xlink:href="#x1D706"/></g><g transform="matrix(.017,-0,0,-.017,72.917,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,79.615,12.162)"><use xlink:href="#x1D434"/></g><g transform="matrix(.017,-0,0,-.017,92.075,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,98.789,12.162)"><use xlink:href="#x1D435"/></g><g transform="matrix(.017,-0,0,-.017,109.278,12.162)"><use xlink:href="#x29"/></g> </svg> of multivalent analytic functions with negative coefficients in the open unit disk. Some modified Hadamard products, integral transforms, and the partial sums of functions belonging to these classes are studied.

Sandwich Results for Certain Subclasses of Multivalent Analytic Functions Defined by Srivastava-Attiya Operator

In this paper, we obtain some applications of first order differential subordination and superordination results involving the operator for certain normalized p-valent analytic functions associated with that operator.

Neighborhoods of Certain Multivalently Analytic Functions

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient bounds, distortion inequalities, and other interesting properties and characteristics for functions belonging to the classes introduced here.

On a Certain Subclass of Multivalent Analytic Functions Defined by the Liu-Owa Operator

By making use of the principle of subordination between analytic functions, we introduce a certain subclass of multivalent analytic functions defined by the Liu-Owa operator. Results such as subordination and superordination properties, convolution properties, distortion theorems and inequality properties, are proved.

Some Properties of Certain Multivalent Analytic Functions Involving the Cătas Operator

We introduce a certain subclass of multivalent analytic functions by making use of the principle of subordination between these functions and Cătas operator. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and sufficient conditions for multivalent starlikeness are provide. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

Some properties of a certain subclass of multivalent analytic functions involving the Liu–Owa operator

By making use of the principle of subordination connecting analytic functions and the Liu–Owa operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved.

Certain subclasses of multivalent analytic functions defined by multiplier transforms

By making use of the principle of subordination between analytic functions and a family of multiplier transforms, we introduce and investigate some new subclasses of multivalent analytic functions. Such results as inclusion relationships, subordination and superordination properties, integral-preserving properties, argument estimates and convolution properties are proved.

ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS INVOLVING A GENERALIZED DIFFERENTIAL OPERATOR

Making use of a generalized difierential operator we intro- duce some new subclasses of multivalent analytic functions in the open unit disk and investigate their inclusion relationships. Some integral pre- serving properties of these subclasses are also discussed.

Certain subclasses of multivalent analytic functions involving the generalized Srivastava–Attiya operator

By making use of the principle of subordination between analytic functions and the generalized Srivastava–Attiya operator, we introduce and investigate some new subclasses of multivalent analytic functions. Such results as inclusion relationships and integral-preserving properties involving these subclasses are proved. Several subordination and superordination results associated with this operator are also derived.

Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure

Making use of the familiar convolution structure of analytic functions, in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here.

Neighborhoods of a Certain Family of Multivalent Functions Defined by Using a Fractional Derivative Operator

Making use of a fractional derivative operator, we introduce and investigate two new classes $K_{j}(p,\lambda ,b,\beta )$ and $L_{j}(p,\lambda ,b,\beta ,\mu )$ of multivalently analytic functions of complex order. In this paper we obtain the coefficient estimates and inclusion relationships involving the $(j,\delta )-$ neighborhood of various subclasses of multivalently analytic functions of complex order.

Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator

By making use of the principle of subordination between analytic functions and the Cho–Kwon–Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

Properties of certain subclasses of multivalent analytic functions involving the Dziok–Srivastava operator

In this paper, the authors introduce and investigate the various properties of two subclasses S p , k l , m ( α 1 ; h ) and K p , k l , m ( α 1 ; h ) of multivalent analytic functions. In particular, some sufficient conditions for the subclass S p , k l , m α 1 ; 1 + z 1 - z of analytic functions, argument properties, the growth theorems, distortion theorems and covering theorems are obtained. Some of the results, presented in this paper, would generalize and extend the corresponding results of earlier authors.

Some subclasses of multivalent analytic functions involving the Dziok–Srivastava operator

In the present paper, we introduce and investigate some new subclasses of multivalent analytic functions involving the Dziok–Srivastava operator. Such results as inclusion relationships, integral representations, convolution properties and integral-preserving properties for these function classes are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

Some New Inclusion and Neighborhood Properties for Certain Multivalent Function Classes Associated with the Convolution Structure

We use the familiar convolution structure of analytic functions to introduce two new subclasses of multivalently analytic functions of complex order, and prove several inclusion relationships associated with the -neighborhoods for these subclasses. Some interesting consequences of these results are also pointed out.

Classes of multivalent analytic functions involving the Dziok–Srivastava operator

The object of the present paper is to investigate some inclusion relationships and a number of other useful properties of several subclasses of multivalent analytic functions, which are defined here by using the Dziok–Srivastava operator. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

Neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator

In the present investigation, by making use of the familiar concept of neighborhoods of analytic and multivalent functions, we derive coefficient bounds and distortion inequalities, associated inclusion relations for the ( n , δ ) -neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which are defined by means of a certain nonhomogenous differential equation. Several special cases of the main results are mentioned.

Certain properties arising from some inequalities concerning subclasses of multivalently analytic functions

In this investigation we prove several theorems involving a general class of multivalently analytic functions in the open unit disk. The applications of the main results are also considered for multivalently starlike functions and multivalenly convex functions. Mathematics subject classification (2000): 30C45, 30A10.