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The estimates for the second Hankel determinant a 2 a 4 − a 3 2 of the analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯  , for which either z f ′ ( z ) / f ( z ) or 1 + z f ″ ( z ) / f ′ ( z ) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained. MSC: 30C45, 30C80.

Purpose This study proposes the best clustering method(s) for different distance measures under two different conditions using the cophenetic correlation coefficient. Methods In the first one, the data has multivariate standard normal distribution without outliers for n = 10 , 50 , 100 and the second one is with outliers (5%) for n = 10 , 50 , 100 . The proposed method is applied to simulated multivariate normal data via MATLAB software. Results According the results of simulation the Average (especially for n = 10 ) and Centroid (especially for n = 50 and n = 100 ) methods are recommended at both conditions. Conclusions This study hopes to contribute to literature for making better decisions on selection of appropriate cluster methods by using subgroup sizes, variable numbers, subgroup means and variances.

In recent years the artificial intelligence has been developed rapidly since it can be applied easily to several areas like medical diagnosis, engineering and economics, among others. In this study we have devised a soft expert system (SES) as a prediction system for prostate cancer by using the prostate specific antigen (PSA), prostate volume (PV) and age factors of patients based on fuzzy sets and soft sets and have calculated the patients’ prostate cancer risk. Our data set has been provided by the Department of Urology, Meram Medical Faculty in Necmettin Erbakan University, Konya, Turkey.

Recently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: ∏ 1 = ∏ 1 ( G ) = ∏ v ∈ V ( G ) d G ( v ) 2 , ∏ 2 = ∏ 2 ( G ) = ∏ u v ∈ E ( G ) d G ( u ) d G ( v ) . These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs. MSC: 05C05, 05C90, 05C07.

Some new results related to the left-hand side of the Hermite-Hadamard type inequalities for the class of functions whose second derivatives at certain powers are s -convex functions in the second sense are obtained. Also, some applications to special means of real numbers are provided. MSC: 26A51, 26D15.

Purpose When a clustering problem is encountered, the researcher must be aware that choosing an incorrect clustering method and distance measure may significantly affect the results of the analysis. The purpose of this study is to determine the best clustering method and distance measure in cluster analysis and to cluster the regions of Turkey on the basis of this result. Methods In hierarchical clustering, there are several clustering methods and distance measures. For comparison of the clustering methods and distance measures, Rand’s C statistic is one of the best methods. Rand’s comparative statistic C takes on values from 0.0 to 1.0 inclusive that may be used to compare two resultant clusterings produced by applying clustering methods to a data set with unknown structure or to assess the performance of a clustering method on a data set with known structure. Results In this study, the seven regions of Turkey are clustered by all the clustering methods and distance measures. Related with the social and economic indicators, the final cluster number is taken as three. Then, according to Rand’s C statistics, all possible pairs of distance measures for all clustering methods in hierarchical clustering are compared, and the results are given in the related tables. Conclusions According to the results of all possible comparisons, Ward’s method is found to be the best among others, and final clustering of the regions is applied according to Ward’s clustering measure.

Let − 1 ≤ B < A ≤ 1 . The condition on β is determined so that 1 + β z p ′ ( z ) / p k ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) ( − 1 < k ≤ 3 ) implies p ( z ) ≺ 1 + z . Similarly, the condition on β is determined so that 1 + β z p ′ ( z ) / p n ( z ) or p ( z ) + β z p ′ ( z ) / p n ( z ) ≺ 1 + z ( n = 0 , 1 , 2 ) implies p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) or 1 + z . In addition to that, the condition on β is derived so that p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) when p ( z ) + β z p ′ ( z ) / p ( z ) ≺ 1 + z . A few more problems of the similar flavor are also considered. MSC: 30C80, 30C45.

The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the q -integers. The q -analogues of well-known formulas are derived. The q -analogue of the Srivastava-Pintér addition theorem is obtained. We give new identities involving q -Bernstein polynomials.

In this paper, we introduce and investigate two new subclasses H σ μ ( λ , φ ) and M σ γ ( λ , μ , φ ) of Ma-Minda bi-univalent functions defined by using subordination in the open unit disk D = { z ∈ C : | z | < 1 } . For functions belonging to these new subclasses, we obtain estimates for the initial coefficients | a 2 | and | a 3 | . The results presented in this paper would generalize those in related works of several earlier authors. MSC: 30C45, 30C80.

In this paper, we discuss the existence of positive solutions for nonlocal q -integral boundary value problems of fractional q -difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem, some existence results of positive solutions are obtained. In addition, some examples to illustrate our results are given. MSC: 39A13, 34B18, 34A08.